The starter relay bypasses the normal fuel pump wiring when the key is in the start position. You typically have enough pressure to run for a few seconds if the normal run position wiring is faulty.
Have a look at the ballast resisitor, jump it for a test, it really doesn´t do much anyway. Have a look at the female spade connectors in the bottom of the relay block. There is usually a bunch of die electric grease under the relays and under the relay block. Pull the plastic hold downs for the relay block and turn it over (carefull of the wiring) and look at the under side, mine was really nasty and corroded. A good cleaning with brake cleaner and/or contact cleaner seemed to help.
If you can find a BMW or Mercedes at the you pick auto wreckers, pull the OEM (Bosch) relays, they are really superior to most others. The OEM (not replacement quality) relays, have a really good coating ont he contact surface that outlasts most every other relay, I´ve tried.
There is a wiring recall, for the relay block (fuel pump relay) on the late 80`s XJ´s (forget exactly which years). But I´ve seen them wired both ways, with no noticeable difference in fuel pump function.