We extend previous works on the study of a particle subject to a three-dimensional spherical singular potential including a $$\delta $$ δ –$$\delta '$$ δ ′ contact interaction. In this case, to have a more realistic model, we add a Coulombic term to a finite well and a radial $$\delta $$ δ –$$\delta '$$ δ ′ contact interaction just at the edge of the well, which is where the surface of the nucleus would be. We first prove that the we are able to define the contact potential by matching conditions for the radial function, fixing a self-adjoint extension of the non-singular Hamiltonian. With these matching conditions, we are able to find analytic solutions of the wave function and focus the analysis on the bound state structure characterizing and computing the number of bound states. For this approximation for a mean-field Woods–Saxon model, the Coulombic term enables us to complete the previous study for neutrons analyzing the proton energy levels in some doubly magic nuclei. In particular, we find the appropriate $$\delta '$$ δ ′ contribution fitting the available data for the neutron- and proton-level schemes of the nuclei $${}^{{208}}$$ 208 Pb, $${}^{{40}}$$ 40 Ca, and $${}^{{16}}$$ 16 O.