How can a marginal connection become "good" by just applying a little more voltage?
Congratulations, you've rediscovered the surprising electrical properties of a loose connection, which have historically played a major role in radio engineering.
In the 1900s, this phenomenon was used to make the first RF detectors, known as a coherer, and numerous variants have been made. The most basic coherer has a loose metal connection, and when an RF signal is received, the voltage across the marginal connection causes the connection to become good, and completes the circuit, thus detecting the signal. Simultaneously, it also means that the coherer is unable to detect a new RF pulse anymore, and it's needed to be &quot;decohered&quot; by mechanical shock, making the connection loose again.
TL;DR: Even after 100 years, there is no universally accepted mechanism on how a coherer works. What is generally known is that the marginal contact is a metal-oxide-metal connection, and applying a voltage, even less than 1 V, is effective to break this oxidation layer between the metal contacts. Proposed mechanisms include electrostatic force, mechanical force from localized heating, micro-welding, or semiconductor physics. The specific mechanism may vary from material to material.

The following selective quotes from the book Early Radio Wave Detectors by Vivian J. Phillips (1980) offered some explanation. The book is available at the Internet Archive. For more information, you may want to read the book. It's a fun read!

Chapter 3: Coherers
The coherer was perhaps the most important of all the early radio wave detectors, and it was used in many different forms. It made use of a phenomenon which occurs in a poor electrical contact, the sort of contact which the engineer of today would call a 'dry joint'. Such an imperfect contact between two conductors normally exhibits a very high electrical resistance due, in large part, to the thin film of oxide which exists between the two metals. When an alternating or direct voltage is applied between the conductors this resistance decreases quite markedly. A voltage of a few tenths of a volt is often quite sufficient to produce the effect.
To give some idea of the magnitudes of the resistance changes involved we may quote the results of an experiment carried out by Professor Edouard Branly, a pioneer in the application of this effect to radio reception. Two oxidised copper rods lying in loose contact showed between them a resistance of 80,000 Ω, after the application of a voltage this fell to 7 Ω. Another early experimenter named Von Lang reported a change from infinity to 380 Ω. Many poor contacts exist in a loosely packed mass of metal filings, and these also exhibit a similar drop in resistance on application of a voltage.
[...]
During the period when coherers were being used there was considerable discussion and disagreement as to how exactly this resistance change came about. In fact, the phenomenon was never satisfactorily explained at that time because other, better devices superseded them and interest was lost before the matter was resolved. We shall return to this fundamental question later, but for the moment we shall simply look at some of the devices in which the phenomenon of coherence was used.
One point needs to be clarified before we proceed further. With almost all these devices the low-resistance condition persists after the coherence has taken place. Once a signal has been received, the device coheres, and unless something is done about it, it is then unable to respond to the arrival of further signals. It can, however, be restored to the high-resistance (sensitive) condition by mechanically shaking or tapping it, and practical systems using coherers incorporated some means of restoring the sensitivity in this way in order to prepare for the reception of further signals. The various methods of effecting this restoration will be considered later, but first we shall examine the coherers themselves in some detail.
It will be convenient to consider first those devices in which only one, or very few, poor contacts were used, and to look at the multicontact filings coherers afterwards.
[...]
Coherers took many different forms - almost as many as the number  of people who used them, in fact. Let us look at some typical examples.




And on its principle of operation...

Before leaving the subject of coherers we must return for a moment, as promised, to say something about how they work; i.e. about the physical effects on which they depend. At the time when they were widely used (broadly speaking, around the turn of the century) there were several theories to explain how the sudden drop in resistance came about. Some were rather vague and spoke of such things as 'molecular rearrangement', 'the electric discharge filling up the intermolecular
spaces', or 'condensing air'. Setting aside such intangible ideas, there seemed to be three main theories. The first assumed that normally the filings or metal electrodes lie in loose contact with one another, but when a voltage is applied across them they experience electrostatic forces which cause them to come together and form chains. Each particle is now pressed into close contact with its neighbours thus establishing a good metal-to-metal contact. Some writers claimed to have seen movements of this sort when the filings were examined under a microscope, and claimed also to have seen sparks jumping from partide to particle' and the formation of chains glowing red-hot within the mass of filings.' Others were very sceptical, arguing that it was impossible to conceive of the production of the large forces necessary to move filings around under the influence of the very small voltages to which the coherer was normally subjected.' Most coherers were, after all, able to operate quite successfully with voltages of only a few tenths of a volt.
It was generally realised that the insulating film of oxide which formed on the surface of the contacts or the filings was of paramount importance in producing a satisfactory device.' One inventor even proposed situating his coherer, which consisted of a contact between copper and lead, over a flame so as to maintain this all-im- portant film. Unoxidised particles were found to make poor coherers, yet on the other hand very heavy oxidation also made for insensitivity and sluggish action. It was this fact that, led Marconi and others to evacuate their coherers of air in order to prevent continuing oxidation which gradually reduced their effectiveness. Others immersed their filings in oil for the same reason. It is also interesting to note that satisfactory coherers were made with particles which had sulphide coatings, or were even covered with wax or resin films to provide a thin insulating layer. As previously mentioned, the noble metals which tarnished slowly were generally found to be unreliable in action. Recognising the extreme importance of this surface layer, and bearing in mind that fact that the Sanskrit word for skin is 'twach', Professor Chunder Bose' suggested that the coherence phenomenon should be called the 'electric touch', a delightful idea which, alas, never caught on.
The second theory to explain coherence was that the mechanical forces caused rupture of the very thin oxide film on the particles followed by welding together at the minute contact points thus created, a variant on this theme was that a tiny hole was formed in the insulating layer, followed by evaporation of a tiny amount of metal which coated the inside of the hole forming a conducting bridge across the insulation. The two theories mentioned so far are not incompatible in that breakdown of the insulation could occur after the formation of particles into chains.
The third theory was that the resistance drop was purely thermal in origin. Most metals have a positive coefficient of change of resistance with temperature — i.e. when you heat them their resistance rises. It was noticed that many of the metals which were satisfactory for use in coherers had oxides which possessed negative resistance coefficients. It was thought that localised heating at the points of contact led to a decrease in the resistance of the oxide film at those points. In certain cases this could lead to thermal runaway where decreasing resistance allowed more current to flow, which in turn decreased the resistance still further and so on, ending with a large flow of current which could weld the particles together.
Argument raged at the time as to which of these theories was correct, but other detectors came along and coherers fell into disuse before the matter could be finally resolved. There have been some modern investigations into the subject. Suffice it to say here that depending on the conditions in any particular case, and especially on the thickness of the oxide coating, all three mechanisms can play a part as the present author's own experiments have shown.

The Wikipedia article Coherer says the exact mechanism can involve deep physics.

Coherence of particles by radio waves is an obscure phenomenon that is not well understood even today. Recent experiments with particle coherers seem to have confirmed the hypothesis that the particles cohere by a micro-weld phenomenon caused by radio frequency electricity flowing across the small contact area between particles. The underlying principle of so-called &quot;imperfect contact&quot; coherers is also not well understood, but may involve a kind of tunneling of charge carriers across an imperfect junction between conductors.

Falcona, Eric; Bernard Castaing (April 2005). &quot;Electrical conductivity in granular media and Branly's coherer: A simple experiment&quot; (PDF). American Journal of Physics. USA: American Association of Physics Teachers. 73 (4): 302–306. arXiv:cond-mat/0407773. Bibcode:2005AmJPh..73..302F. doi:10.1119/1.1848114


And in the cited paper, it determined that, in the particular case of a metallic granular media, the micro-welding effect is responsible for the dramatic decrease in resistance.

Although these electrical transport phenomena in metallic granular media were involved in the first wireless radio transmission near 1900, they still are not well understood. Several possible processes at the contact scale have  been invoked without a clear verification: electrical breakdown of the oxide layers on grains, the modified tunnel effect through the metal-oxide/semiconductor metal junction, the attraction of grains by molecular or electrostatic forces, and local welding of micro-contacts by a Joule heating effect. Global process of percolation within the grain assembly also has been invoked.
[...] We have reported the observation of electrical transport through a chain of oxidized metallic beads under an applied static force. A transition from an insulating to a conducting state is observed as the applied current is increased. The U–I characteristics are nonlinear, hysteretic, and saturate   to a low  voltage per contact (0.4 V). From this simple experiment, we have shown that the transition triggered by the saturation voltage arises from an electro-thermal coupling in the vicinity of the micro-contacts between each bead. The current flowing through these spots generates local heating which leads to an increase of their contact areas, and thus enhances their conduction. This current-induced temperature rise~up to 1050 °C, results in the microwelding of contacts~even for a voltage as low as 0.4 V! Based on this self-regulated temperature mechanism, an analytical expression for the nonlinear U–I reverse trajectory was derived, and was found to be in good agreement with the data.

I found another master thesis, Coherers, a review by Thomas Mark Cuff that speculates there may be some deep semiconductor physics going on here.

As a result of the historical review, it became clear that the coherer evolved directly into the MOM (Metal-Oxide-Metal) ‘diode’ and, by only a slightly more circuitous route, it appeared as the forerunner to the STM (Scanning  Tunneling  Microscope). The MOM ‘diode’, besides being a progeny of the coherer, has something else in common with thecoherer, no generally  accepted explanation of how it works.