I do remember in my parents house, hunting for why a socket did not work, I found my dad had fitted a fused connection unit (FCU), feeding another FCU and one was under the counted hard to see.
So I can see how hard it is to find the problems sometimes.
The worry is if part of a ring final and the fault is breaking the ring. This can cause an overload in one leg, so unless we know that is not the case, we do tend to spend a lot of time inspecting and testing to make sure no danger.
So as
@pete01 says, fixing the socket is best. But you asked about use of an extension lead, and there is no real problem other than what I have already said, and the trip hazard.
I would remove the socket, and see how many cables feed it, if only one, then not likely part of a ring final, if two then more worrying, and you need to look further. Since a freezer, it was common to fit switches to sockets supplying freezers so it could be turned off without moving the unit, where the unit its self was in front of the socket, to get 20 amp rated switches, it was common to use grid switches, these also allowed on to get switches with a legion on them.

It could also combine a fuse holder and switch. Or a row of switches
supplying other items which are in front of their sockets, it was considered important with washing machines, as there was a fault where the weights we made slightly undersize, and the fact was masked by flashing on the concrete, when this flashing powered off, the washing machine could vibrate around the room, so wanted a way to turn them off, without touching it. Today they have out of balance sensors so auto turn off.