Old monitor’s internal power supply works only when heated from the outside...fixable?
I see at least two things:

First is the capacitor that's already been pointed out to you.  

That's this one:



It can cause many kinds of problems.  It needs to be replaced, no matter what else may be wrong.

The second thing I see is this joint on the back of the board:



It looks to me like the left pin of C810 has come unstuck.  There's what looks like a ring around the pin.  It might be just a shadow, but I think it's a gap between the pin and the pad.

I'd probably touch up all the joints just to be safe.  That kind of thing will respond to heat and wiggling, so it might be your culprit.