For this integral:
$$\int_1^0 \pmb{H}_0\left(\sqrt{1-z^2} z\right) \, dz=\frac{2 \, _2F_3\left(1,1;\frac{5}{4},\frac{3}{2},\frac{7}{4};-\frac{1}{16}\right)}{3 \pi }$$
we use a Mellin Transform to solve:
InverseMellinTransform[
Integrate[
MellinTransform[StruveH[0, A*Sqrt[1 - z^2] z], A, s], {z, 0, 1},
Assumptions -> s < 1], s, A] /. A -> 1 // FullSimplify
(*(2 HypergeometricPFQ[{1, 1}, {5/4, 3/2, 7/4}, -(1/16)])/(3 \[Pi])*)
Regards M.I.