ok, so are you asking if u can cut the other 2 wires to/from the relay? caus if so u cant as u need all 4 wires to run the relay/pump
u probably know this but:
the trigger wire from the std loom tells the relay when to open/close the circuit. when the relay see's power from that trigger, it closes the circuit and u get power through the relay. if u dont have power in, then u wont have power out. if u also dont have the relay ground connected, the circuit will be open and u wont get power out.