Internal damp on gable end wall

I think I would look at your ventilation. Gable end walls are notorious for being cold especially if it faces into the prevailing weather direction. Since your kitchen is against this wall and is a major source of moisture as a result of cooking the likelihood your damp/black mould is a result of condensation. The cure is keep the room warmer and ventilate when cooking, if you do not have a ventilator than open a window. To remove existing black mould bleach does quite a good job.

Richard

Although of course it is always worth looking for possible penetrating damp problems from the outside, leaking gutters or downpipes, cracked rendering etc.

NT

Yes, definitely first thing to look at before looking for leaks etc. You probably need a cooker extract hood (piping the damp air outside and with arrangements to let "replacement" air back in.)

Black mould is hard to get rid of when established. Lots of bleach!

You can't put cupboards against external walls unless the walls are well insulated. Otherwise, the cupboards end up forming the insulation, meaning the wall behind will be below the dew point, and condensation is inevitable. Wardrobes are the worst case, because the contents often form a better insulation than anything in the wall, meaning it will be cold at the back and condensation and damp clothes is inevitable.

You might investigate if cavity wall insulation can be installed. Alternatively, internal or external wall insulation might be an option.

If it's one unit/cupboard, you could remove it and put a thin insulation board behind it. The insulation board needs to be sealed to the wall all the way around and have a non-permeable inner surface (and edges) at least. However, I would think carefully before fitting a sheet of PIR in a kitchen, particularly behind anything like an oven. It will make any fire in the kitchen lethal in the whole house because it gives off cyanide when heated in a fire.