The coupling between mixed convection and surface radiation in a square cavity, with a vertical wall heated and driven, and the opposite one subjected to a non-uniform temperature that exhibits a sinusoidal spatial variation is studied numerically using the lattice Boltzmann method and air as working fluid. The numerical code is validated against experimental and numerical results available in the literature. The parameters governing the problem are the emissivity of the walls varied from 0 to 1 and the Richardson number varied from 0.01 to 500. The latter is varied either through the Grashof number (the Reynolds number being fixed at 100) or through the Reynolds number (the Grashof number being fixed at or ). The results obtained show significant effects of the Richardson number on the overall structure of the flow and heat transfer characteristics. It is also shown that the contribution of radiation to the total heat transfer is not negligible even at low Richardson numbers and this contribution is reinforced by increasing the latter parameter. Useful correlations are presented for the total Nusselt number versus the Richardson number (in the range 1-500) and the emissivity of the walls (in the range 0-1).