Does your primer pump have a white plastic cap that you unscrew then pump? If so that is the old type and an easy R+R, they can leak air. Also was this a good running vehicle that this developed all of a sudden or a project type vehicle that you do not have much history on?

Maybe give a little more history, as it could be faulty fuel gauge and you are just low on fuel, clogging tank strainer, could also be a plugged air filter and you are associating using the pump to restart as a fuel issue.