It depends on ownership and what you're allowed to do under your rental agreement, but this was the solution on our house. It's stick built 2x4 construction, you didn't say what wall construction is used, so it may not apply. Same situation, north facing, cold corner, black mold growing in the plasterboard surface and paint about 4 inches back from the corner on both walls, looked like someone used a paint roller to apply it.
I stripped the plasterboard off the wall to the stud closest to four feet from the corner. The 70's insulation was mouse power R-6 pretending to be something like R-13 but which didn't completely fill the air space.
After clearing everything out of the stud bays and bleaching the minor surface mold that was on the exterior sheeting, I examined what I had to deal with.
The most egregious problem was the empty box created where the two walls came together. We're talking about uninsulated dead air space that is open to the outside atmosphere and sealed by the exterior trim. It was a source for both draft and cold. You could feel the air flowing into and out of the room. I took a wood knife I had and opened up the spaces between the 2x4s that enclose this empty space so I could get a nozzle straw in from floor to ceiling so I could fill it with Urethane foam. You don't want to do the whole space all at once, urethane foam in quantity develops a skin, massive internal voids and blows itself out of the containing space in massive mushrooms that waste the material. This is why you want a slot that gives freedom of movement for the nozzle straw. After about six passes and a couple days for full expansion and cure of each foam application, the void was completely filled with minimum waste. No more cold vented dead air space or drafts!
I got R-15 which was the highest R rating available for 2x4 studded walls (R-19 is standard for 2x6) and filled the stud bays completely with no voids. Over the top of this went 6 mil plastic for vapor barrier so no internal moisture would intrude. Put the plasterboard back, taped the joins, copied the simple skim coat, primed and repainted the two walls to match.
After the project, the corner is actually warmer than the rest of the wall because it has working insulation with no empty, uninsulated and unsealed air spaces. No more mold!!! My only regret is I probably should have done the whole whole room on the reinsulation.