No start condition, cranks over.

I pulled the fuel relay, jumped the relay and the puel pump runs, but in crappy kind of way - kind of gurgles, not a smooth usual fuel pump kind of purring. When turning the key, the car still does not engage and run while the relay is jumped.
When I remove the jumper wire and crank the engine, the car turns over and kind of wants to start for a millisecond and dies. This does not occur when the relay jumper wire is in place.

When I put the fuel pump relay in, the car fuel pump does nothing. No noise or any signs of running.

So i am assuming it's s fuel issue. I'm thinking, specifically a fuel pump issue, but wanted some opinions before I buy a fuel pump.

Thanks!!

Dwayne.