The fittings are a bit Heath Robinson Stewart. It's a dilemma because a new one might not be much better. Have you tried a new bulb? The prongs will be a bit longer and it might just hold better.
I’ve tried three different lamps and they all do the same
Work when pushed in firm but as soon as I release them they stop working
Bit of a puzzler as I can’t understand why it should just suddenly stop working - there doesn’t seem to be anything broken