The aim of this paper is to find the exact solutions of the Schrödinger equation for the Manning–Rosen plus Graz separable potential through two different approaches to the problem. We express the irregular/Jost and physical solutions in terms of the special functions of mathematical physics. Numerical results of the phase shifts are obtained by utilising the properties of the Jost function and Fredholm determinants for nucleon–nucleon and nucleon–nucleus systems. The results obtained are in good agreement with earlier works.