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) *)
Edited 20.07.2023:
InverseLaplaceTransform[Hypergeometric2F1[1, 1/p, 1 + 1/p, s], s, t] ==
InverseLaplaceTransform[FoxHReduce[Hypergeometric2F1[1, 1/p, 1 + 1/p, s], s], s, t] == (
Gamma[1 + 1/p] Inactive[
FoxH][{{{0, 1}, {(-1 + p)/p, 1}}, {}}, {{{0, 1}}, {{-(1/p),
1}, {1, -1}}}, -(1/t)])/(
t Gamma[1/p]) == ((-(1/t))^(-1/p) Gamma[(-1 + p)/p, -t])/(
p t) == -(ExpIntegralE[1/p, -t]/p)
. Inverse Laplace Transform of FoxH function I use forumlaformula from book on page:51