As Andy said... New Plug AND Socket needed...
These sorts of problems are not uncommon with either the neutral or the live pins,
Often due to any combination of:-
Loose termination in plug..
Poor quality connections between the plug & the socket..
Loose fuse carrier connections..
e.g. some past examples I have encountered...
(1) burnt out live pin... and melting in the back of the socket..
(2) Overheated plug fuse carrier..
(3) Over heating causing the plastic insulation part-way up the pins to melt and distort..
Basically Overloading, loose connections, natural ageing and wear and tear.. etc.. etc.. could all be a mixture of your problem here! Which all comes back to Andy's answer.. "New Plug AND Socket needed"