  In this work we consider sub-leading $O(Z^2\alpha^3)$ corrections to coherent
elastic neutrino nucleus scattering (CEvNS). These corrections are not
negligible by power counting since nuclei with large coherent cross sections
have sizeable nuclear charges e.g.\ $Z\alpha \sim 0.4$. We find that the
corrections are much smaller than naive power counting in $Z\alpha$ would
suggest and that Coulomb corrections do not substantially alter predictions for
CEvNS in the Standard Model. We comment on similarities to older literature on
mesonic and muonic atoms.
