There is a wiring problem between the relay and the fuel pump. A gray wire goes from the oil pressure switch, joins a gray wire from the fuel pump relay, and goes to the fuel pump. The connection could be faulty.
Read ohms, on the oil pressure switch gray wire, to the fuel pump. It should be less than 1 ohm. Read the ohms from the relay, on the gray wire, to the fuel pump. It should be less than 1 ohms. If you get more ohms on the gray wire, when measured from the relay, but ok when measured from the oil pressure switch, that section of wire from relay to the connection (or, the connection, itself) is faulty (maybe corrosion).
Measure the ohms through the fuel pump to its ground. The resistance shouldn't be more than the resistance of the fuel pump motor. If you have the specs sheet of the fuel pump, it will tell you the ohm value of the motor. If the resistance is higher, check connections and ground point.