Electric motor will only start turning at lower voltages
I'll turn our comments into an answer.


What is actually happening to cause the pulses? Is my power supply overloading then &quot;resetting&quot; once every second?


It's likely that the motor is tripping the current protection circuitry on the power supply when it's starting up. Then the power supply resets and all starts again generating the pulsing effect.

2.Is there something very simple I can add into the circuit to allow it to start at higher voltages (a momentary current spike)?
3.Is there something very simple I can add into the circuit, perhaps a better power supply, to allow it to run under sustained load (a constant current increase)?

In his comments, WhatRoughBeast suggests you get a bigger power supply.
I've expecienced that once and what I did to work around it was to replace the wall-wart by an equivalent set of AA bateries. They usually let you draw a few amps at start up without any limiting circuitry to get in the way. In your case, 4 AAs in an appropriate battery holder will do the trick.