Download the service manual set for your vehicle from the Sticky thread section of the Engine forum.
Follow the diagnostic steps for Code 53.
The fuel pump and the gas-gauge rheostat are in the same assembly inside the fuel tank, but low voltage to the fuel pump relay is almost certainly NOT related to the fuel gauge inaccuracy.
If the gas gauge reads higher than the true level, (very common) it's probably due to high resistance (corrosion) in the wire harness between the gauge and the sending unit in the tank; or between the sending unit and ground.
If the gas gauge reads lower than the true level (not so common) it may be a problem within the gas tank (bent float arm, float that's gotten heavy)
Fuel gauge diagnosis will also be in the service manual.