I've recently had a weak sounding pump and refurbished mine back to full health, here's a few things I noticed:
1) I did see the 0.3v and the plug when I was expecting the full 12. This may be (as already mentioned) the cooling control unit detecting short-to-ground or open circuit - don't know if VCDS would show anything, sadly I didn't check. I solved this by idling for about 2 mins with the refurbished pump plugged in, it worked again.
2) I found had originally diagnosed this as a wiring
problem - I could wiggle the wires near the pump and in certain orientations the pump would nearly die. This was actually moving the rear bearing of the pump slightly and causing the (degraded) motor brush to disconnect - so a pump
problem, not a wiring
problem
3) I took apart both ends of the pump. It's not completely necessary but it does allow you to get carbon out of both ends. The fix was simply to remove all the carbon and crap and make sure both brushes moved really freely in their runners. I had to use two loops of thread to hold the brushes away from the shaft during reassembly which was fiddly, but not possible.