Sounds like a fueling problem. I had something similar ages ago, for me it turned out to be the fuel pump under the body, well, ish ish. The wiring that went to the pump (under the body) had worn through and was causing an intermittent power feed. Fuel, no fuel, cut out, start again, not start, start, cut out!! Did me nut in and took some finding as I went down all the other routes first, turning the key and hearing the pump prime threw me off too, but got there in the end.
If that's not your exact problem, still sounds like a fueling issue!
Good luck!!