When the key is turned to the START position the fuel pump relay is closed by a direct connection from the key switch/starter solenoid. When the key is released to the RUN position the pump relay is under the control of the ECU (through yet another relay). Look for a fault in one or both relays, or in the ECU. In this case, when the key is released to RUN, the fuel pump stops, and you are probably hearing a relay clicking.