that is a chance that is the problem
first, lets test the battery. any shop or auto parts store will load test the battery for free, and that will eliminate the any issues with the battery. a load test will put a load on the battery, simulating a starter, and read the voltage. the battery needs to have at least 10 volts for 7 seconds, and needs to do it twice. if the battery is good, then there is something wrong with the fuel pump(maybe)
if the battery is bad, it can trigger a FI light because battery voltage is below thresholdm or below needed voltage to run the fuel injection system.
you need a multimeter to test the voltage going to the fuel pump. if you get at least 10 or more volts to the fuel pump, and it is not working, then that is your fault code. a bad not working fuel pump will trigger a FI light. if the fuel pump is clogged, but still running, it will not trigger a code. with a multimeter, on volts DC in the under 20v range(if it is not autoranging) with the key and on switch on you should get the battery voltage at the fuel pump.
if you do and the pump is not working that is the bad part.
if you get 0 volts or low volts, then the fuel pump relay might be at fault, and is causing a problem sending the right voltage through the relay.
test the battery first, then the voltage at the fuel pump next