We have studied the bound-state solutions of the spin and pseudospin limits of the Dirac equation with the generalised Morse potential and a class of Yukawa potential using the Nikiforov–Uvarov method and an appropriate approximation scheme. The energy eigenvalues and the corresponding normalised eigenfunction of the non-relativistic limits of the spin symmetry was obtained. By adjusting some potential parameters, six special potentials, namely, the generalised Morse, a class of Yukawa, Hellmann, inversely quadratic Yukawa, Hulthen and Coulomb potentials to which the generalised Morse potential plus a class of Yukawa potential (GMP + CYP) reduces to, were evaluated. The deduced corresponding energies of these special potentials were found to be in excellent agreement with those in the existing literature. We also studied and compared the energy spectra of the GMP + CYP with those of the special potentials with respect to internuclear distance. Moreover, we investigated and compared the behaviour of some selected diatomic molecules, when subjected to the (GMP + CYP) potential. Finally, the expectation values of some useful physical observables were deduced using the Hellmann–Feynman theorem.