when the key is turned on, the computer sends a flash signal to run the pump, but only for 1 or 2 seconds then it stops. that is normal. Once the engine is running, it will then kick on all the time, especially if ur running a two line fuel system-- supply and return. i have diagnosed this issue a 100x and the best way to isolate or eliminate the issue is to test the PSI at the fuel rail. Otherwise, you can fret over many things that may not be relevant to the fix.
do you have in tank module or external Airtex style pump?