The study on the mass splittings of the mesons with the same structure but different spin-and orbit-quantum numbers is one of the important methods for checking the efficiency of potential models. In previous calculations for quark potential models, the splitting between - is easily obtained while that of the c-J/ is however too small to meet the experimental results. In this paper, the third term of the complete Breit quark potential in the momentum space is regularized twice by applying the form factor 2/(q2+2), and the other terms except the first term of the Coulombic potential and the seventh term of the constant potential are regularized once. The mass splittings are calculated by using these values. Our results indicate that the mass splittings of light mesons -, heavy mesons c-J/, b-(1s), and c0-c1-c2 can meet the experimental results with high accuracy only when the screen mass is expanded to the third-order polynomial with respect to the meson reduced mass r=mr mj/(mr+mj), while the masses of other mesons are improved greatly. An efficient quark potential model is thus described in this paper.