Powerbank voltage drop after while
A power bank's boost circuit, which generates the nominal 5V output voltage from the nominal 3.7V battery voltage, will consume some power even when no load is connected.  Therefore, it must be turned off when not in use if you want the power-bank to be ready with a full charge when you need it.

Different power banks seem to have different strategies for automatically switching the boost circuit on and off, but what you describe seems to fall within the normal range:


Automatically powers up when a load is sensed,
Automatically powers off when the load current has been below some threshold for some amount of time,
Doesn't power up again until the load is completely removed and then re-connected.


The 1.1V that you measured probably 
is needed to sense the presence of a load when the boost circuit is switched off.