In this study, we solve the non-relativistic radial part of the Schrödinger wave equation for the superposition of the Hulthen with spin-orbit plus adjusted Coulomb potential (SHSC) potential using the Nikiforov-Uvarov (NU) method for arbitrary states. The Hulthen with spin-orbit plus adjusted Coulomb (SHSC) potential is the simplest potential field for a nuclear system and has been used to obtain the single particle energy spectrum for both nucleon species orbiting a closed nuclear core. We also obtained in this study the corresponding single particle normalized wave function expressed in terms of the Jacobi polynomial. Besides, we obtained two special cases of the energy spectra for the SHSC potential.