1979 Cadillac Seville mind of his own.

My car has a "mind of his own".
It starts good and then the engine dies after 1-2 seconds.
If I "play around"with the gas pedal it "sort of "keeps running but when I let go of the gaspedal it stops again.
But on other moments it starts and runs smootly, no problems and this happend twice when my mechanic was around to check it.
if I turn the key, right before it starts I hear the "switch" of the"main" fuel pump(on the gas tank) "kick in' so I gues that works.
Could it be the "second"fuel pump, under neath the body, that works only now and then?