Within the formalism of the effective mass approximation, the Schrodinger equation is solved using a Hass variational method for a cubical GaAs/GaAlAs core/shell quantum dot. The impact actuated by the outside electric field, the hydrostatic pressure, the donor impurity off-center, and on-center effect, as well as the confinement geometric on the diamagnetic susceptibility, the polarizability, and the photoionization cross section (PCS), has been analyzed, taking into account the effect of the interaction electro–phonon (e–p). Our numerical results reveal that all of these principal parameters are very subject to the core and shell width. In addition, the impurity position and the electric field have a significant effect on the PCS without hydrostatic influence, especially when the impurity is situated at the center, and the situation becomes less significant in the boundaries of the system. On the other hand, it is found that the impact of both e–p interaction and hydrostatic influence causes a blueshift of the PCS. Interestingly, the polaronic effect is enhanced with the decreasing of the inner quantum dot width.