  We have constructed a perturbation theory to treat interactions that can
include the Coulomb interaction, describing a physical problem that is often
encountered in nuclear physics. The Coulomb part is not treated perturbatively;
the exact solutions are employed. The method is an extension of the results
presented in Hoffmann (2021 J. Math. Phys. 62 032105). It is designed to
calculate phase shifts directly rather than the full form of the wavefunctions
in position space. We present formulas that allow calculation of the phase
shifts to second order in the perturbation. The phase shift results to second
order, for a short-range potential, were compared with the exact solution,
where we found an error of third order in the coupling strength. A different
model, meant as a simple approximation of nuclear scattering of a proton on
Helium-4 and including a Coulomb potential and a spherical well, was
constructed to test the theory. The wavepacket scattering formalism of Hoffmann
(2017 J. Phy. B: At. Mol. Opt. Phys 50 215302), known to give everywhere finite
results, was employed. We found physically acceptable results and a cross
section of the correct order of magnitude.
