Look it's NOT rocket science, a 1 second prime is unlikely to be of any use to anyone, if that's the duration, especially after being left overnight, there is a valve in the pump to stop drain back, but they seriously are not that good, and won't hold pressure overnight, not even a brand new one.
The old relay cannot alter the prime duration, but can be easily(and safely) taken out by putting a wire across terminal 30(battery) and 87(output), the output from the fuel pump relay also feeds the amal valve, so unless you do a bit of rewiring you need a jumper across 30 + 87, but as the load if massively reduced only a thin wire is needed to run the amal valve, there's no point though unless it is causing a problem.
tabetha