We need to solve a suitable exponential form of the position-dependent mass (PDM) Schr(o)dinger equation with a charged particle placed in the Hulthen plus Coulomb-like potential field and under the actions of the exteal magnetic and Aharonov-Bohm (AB) flux fields.The bound state energies and their corresponding wave functions are calculated for the spatially-dependent mass distribution function of interest in physics.A few plots of some numerical results with respect to the energy are shown.