  The relativistic wave equations determine the dynamics of quantum fields in
the context of quantum field theory. One of the conventional tools for dealing
with the relativistic bound-state problem is the Klein-Fock-Gordon equation. In
this work, using a developed scheme, we present how to surmount the centrifugal
part and solve the modified Klein-Fock-Gordon equation for the linear
combination of Hulth\'en and Yukawa potentials. In particular, we show that the
relativistic energy eigenvalues and corresponding radial wave functions are
obtained from supersymmetric quantum mechanics by applying the shape invariance
concept. Here, both scalar potential conditions, which are whether equal and
non-equal to vector potential, are considered in the calculation. The energy
levels and corresponding normalized eigenfunctions are represented as a
recursion relation regarding the Jacobi polynomials for arbitrary $l$ states.
Beyond that, a closed-form of the normalization constant of the wave functions
is found. Furthermore, we state that the energy eigenvalues are quite sensitive
with potential parameters for the quantum states. The non-relativistic and
relativistic results obtained within SUSY QM overlap entirely with the results
obtained by ordinary quantum mechanics, and it displays that the mathematical
implementation of SUSY quantum mechanics is quite perfect.
