Maybe:

    FullSimplify[InverseMellinTransform[InverseLaplaceTransform[
    MellinTransform[Hypergeometric2F1[1, 1/p, 1 + 1/p, s/a], a, q] // 
    FunctionExpand, s, x], q, a] /. a -> 1]

    (*-(ExpIntegralE[1/p, -x]/p) *)