Yeah that looks like condensation on a cold solid wall.
A) Turn your heating up.
B) Open windows (open trickle vents).
C) Don’t dry washing in the house.
D) Make sure your bathroom and kitchen have working extractor fans and use them.
E) Close the bathroom and kitchen doors in/after use with extractor running on.
If that doesn’t work
F) Insulate the wall.
You could put insulated plasterboard on (stick it on with no gap behind).
Or maybe first you could try thermal liner paper e.g. https://www.toolstation.com/wallrock-thermal-liner/p43007
I once had a condensation problem under bay window of an old terrace, I stuck some of that radiator foil bubble stuff on under the window and then lining paper on top. Worked well, but not a solution for a whole wall though.
https://www.toolstation.com/ybs-radiator-reflector-foam/p27189
The cracks are nowt. You should see the cracks in a new build after a few years. Just decoration.