The car should still run if the non return valve is gone but it will be a slag to start as there is no pressure in the fuel system, bypass the relay and allow the pump to run for at least 30 secs and see if it will start then. Or alternatively leave it overnight and see if theres pressure in the system in the morning before you try and start it.

My valve was gone, it did run bit it took about 3+ minutes of solid cranking to start it.