The brief
The A4 had a coolant pump on its way out, showing the signs: a weep around it, the level creeping down, and a faint whine starting. The signs of a failing coolant pump are exactly that, and they're worth acting on before it fails properly and overheats the engine. He brought it in. The coolant pump circulates coolant through the engine and the radiator so the heat goes somewhere, and on this engine the pump and the thermostat come as one module. As the pump ages the seal weeps coolant and the bearing gets rough and noisy, and once it fails properly the engine can't shed heat and overheats. A weeping pump doesn't reseal, and on these you change the module so the thermostat goes with it. So it needed replacing.
The diagnosis
A pressure test on the cooling system pinpointed it, the water pump module was weeping from its seal and losing pressure slowly, with the bearing rough, which is the disappearing coolant and the whine. The radiator, the hoses, the expansion tank and the rest of the system held fine. That's a pump module replacement, with the integrated thermostat and a fresh seal, rather than chasing a weep that's only going to fail harder.
The work
The cooling system was drained, the old water pump and thermostat module removed, and a new genuine Audi-spec module fitted with a fresh seal and the drive belt set back up properly. The system was refilled with the correct Audi coolant, the air bled out the proper way so no pockets were left, and held under pressure to confirm the seals were dry. A road test confirmed the gauge sat steady through traffic and at speed with no overheating and no noise, and the level stayed put.
The outcome
No more coolant loss, the level holding between checks, the gauge steady, the engine warming up on time, and no whine from the bay. The A4 went home with the cooling system circulating properly again. A weeping coolant pump only fails harder, and the failure at the end is an overheat that can take the head gasket with it, so changing the module kept it to a tidy job.
