Hi,
If you are able to turn the fuel pump on by jumping the relay, we need to check things at the relay itself. If you have a different relay with the same part number, switch them to see if it works. If you don't, here is a link that explains how to test a relay:
https://www.2carpros.com/articles/how-to-check-an-electrical-relay-and-wiring-control-circuit
Now, if you look below, I attached a pic of the relay wiring schematic. I need you to first confirm both of the fuses I highlighted. In addition to confirming the fuses are good, make sure there is power to and from them.
If they are good, remove the relay. On the relay there will be 4 numbers, 30, 85, 86, and 87. Pins 30 and 85 are the power supplies. With the relay removed, determine which connector in the box each pin goes to and confirm pins 30 and 85 have power. Pin 85 will only have power with the key on. Pin 30 will have power at all times.
If these two pins have power, the last thing left (based on the idea you were able to jump it) is there is no ground path being provided by the PCM (pin 86). When you turn the key to start the vehicle, the PCM provides a ground to pin 86 completing the circuit on the primary side of the relay. When that happens, an electromagnet is energized closing the switch on the secondary side allowing power to run from pin 30 to 87, powering the pump.
Here is a link you may find helpful:
https://www.2carpros.com/articles/how-to-check-wiring
Also, let me know if you have checked the auto shutdown (ASD circuit as well.
Take care and let me know what you find or if you have other questions.
Joe
See pic below.
Image (Click to make bigger)
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Monday, September 26th, 2022 AT 8:09 PM