  In the theory of electron-phonon superconductivity both the magnitude of the
electron-phonon coupling $\lambda$ as well as the Coulomb pseudopotential
$\mu^*$ are important to determine the transition temperature $T_c$ and other
properties. We calculate corrections to the conventional result for the Coulomb
pseudopotential. Our calculation are based on the Hubbard-Holstein model, where
electron-electron and electron-phonon interactions are local. We develop a
perturbation expansion, which accounts for the important renormalization
effects for the electrons, the phonons, and the electron-phonon vertex. We show
that retardation effects are still operative for higher order corrections, but
less efficient due to a reduction of the effective bandwidth. This can lead to
larger values of the pseudopotential and reduced values of $T_c$. The
conclusions from the perturbative calculations are corroborated up to
intermediate couplings by comparison with non-perturbative dynamical mean-field
results.
