It is well-documented that adding the Hartree-Fock (HF) exchange to the energy functional can alter significantly the band gap of solids with two extremists: pure HF which overestimates band gap and a LDA (or GGA functional) of DFT which underestimates the band gap. This is the main reason behind the good performance of hybrid XC in computing band gap where a portion of HF exchange is used instead of just simply using some exchange functional. The physical rationale here can be seen as partially adding the HF exchange corrects the self-interaction error (electron sees itself!) in the approximate DFT functionals since there is no self-interaction in the HF exchange.

In BJ potential, a Slater potential is used to capture the "exact" exchange effect without the need of computing the HF exchange from 2-e integral (which is very expensive). The mBJ replaced Becke-Russel potential for the Slater potential in BJ potential because these potential give similar results on atoms.

Loosely speaking, the mBJ exchange potential is mimicking the hybrid functionals where a portion of the HF exchange (calculated from the Becke-Russel potential in this case) is used to improve the prediction of band gap. Herein, the mixing parameter c can be assigned as a constant or self-consistently determined (DOI: 10.1103/PhysRevLett.102.226401). It should be noted that mBJ is just an exchange potential and can be used in combination with a LDA correlation potential.