Fuel pump (?) woes

[GreyB5.5]

My petrol passat 2 litre 20v doesn't want to run. I drove it early this morning and thought I noticed a little bit of hesitancy a couple of times, but didn't notice it again and assumed it was my imagination. I continued another 20 minutes without issue. I then parked up for 15 mins while I did some shopping. When I came back it started and ran very briefly and then died. On trying to restart it, I noticed the fuel pump wasn't running. I left it 30 minutes and tried it again. It turned over fine, but would not fire. I could not hear the fuel pump.

I called recovery and they tested for a spark and got it running momentarily on easy start and reached the same conclusion as me - fuel pump.
I've got it home and started checking things, but I don't know what my results mean:
- The fuel pump relay opens and closes with a 9v battery placed across the control terminals. When closed it shows zero resistance across the power terminals (the large blades). That says to me the relay is probably OK.
Now the bits I don't understand...
- When I turn on the ignition with the relay removed, I'm getting about 8.7 volts where the relay plugs in (across the big / high current terminal slots)
- When I plug the relay back in and turn on the ignition, I get a consistent 3.5 volts at the plug that connects to the pump (disconnected) when I turn on the ignition.

I don't understand the low voltages! I'm not drawing any significant current during my tests, so even if a cable is in poor condition with high resistance, surely I should still get 12 volts.

The battery is well charged and shows a consistent 12.3 volts.

Any pointers would be very much appreciated Thank you!