The brief
The Golf's aircon had stopped giving cold air, with loud noises from the system, the compressor cycling on and off, a refrigerant leak, a dash warning, vibration and a burning smell from the vents. He brought it in. That whole list points at the AC compressor. The compressor is the pump that compresses the refrigerant to cool the air before it enters the cabin, running off the engine belt with a clutch. When it fails, the cooling drops off, it gets noisy, it cycles erratically, and a breaking-up compressor can leak and send debris through the rest of the system, which can damage other parts. A compressor doing all of that has had its run.
The diagnosis
A check confirmed it: the compressor wasn't building proper pressure, its internal parts were worn and noisy, and it was leaking. The condenser, the lines and the evaporator checked out, but with a failing compressor the system gets flushed anyway to clear any debris before the new one goes on. That's a compressor replacement, with a receiver-drier and a flush, not something a recharge alone fixes.
The work
The remaining refrigerant was recovered, the drive belt released, and the failed compressor unbolted and removed. The system was flushed to clear any debris, a new genuine VW-spec compressor fitted along with a fresh receiver-drier and new O-rings, the belt set back up, the system pulled down to a long, deep vacuum, and recharged with the correct weight of refrigerant and the right oil charge. A check at the vents confirmed cold air at idle and no noise from the compressor.
The outcome
Cold air at the vents at idle, strong cooling, no noise from the AC, no burning smell, and the charge holding. The Golf went home with the aircon working the way it should. A failing compressor only gets worse and can take the rest of the system down with it, so changing it, flushing the system and recharging properly fixed the cooling and protected the rest of the AC.