The fuel pump turns on when the key is turned on, but loses power once the engine is started. The ignition has been replaced, all relays are good and the vehicle runs fine, it just loses power to the fuel pump once the engine is running. Can anybody shed some light on the subject for me.

Is this a new problem? Does it lose power while your cranking? Or only after it starts?

If it's the latter then check your oil pressure switch output. Some cars have it set so if the engine dies it turns off the pump. If its bad then you won't get power after it is running. Just during ignition "on".