I have a very similar problem with mk5 1991. After trying several things I made a jumper connection with a switch. When the fault occurs I put 12V directly to the pump. I know it is not the most professional solution but the car starts and does not stop. I have discarded that it is a problem of the relay, I changed it, and also of the pump, since in this way it always works.
I suspect that it is a problem with the fuse box or wiring but I have not found adequate electrical diagrams to analyze this fault.
Regards