Goodman GM100 Failure to recycle
If wiggling (which is the correct technical term) can make it stop, or start again, the problem is most likely a broken or mostly broken wire.

The wire can be broken inside of the insulation, and look fine.

I have the broken wire inside the insulation all the time, my cats love my rubber phone charge cord, and will chew it. When I go to charge my phone, the charger will not work sometimes, and i can find a single cat tooth mark that was able to cut the wire.