A two-dimensional numerical study is performed in an effort to understand the fundamental characteristics of hydromagnetic mixed convective transport in a nonisothermally heated vertical lid-driven square enclosure filled with an electrically conducting fluid in the presence of a heat-conducting solid circular cylinder. Additionally, entropy generation due to heat transfer, fluid friction, and magnetic effect is also determined. Simulations are performed for various controlling parameters such as the Richardson number (1 <= Ri <= 10), Hartmann number (0 <= Ha <= 50), Prandtl number (0.02 <= Pr <= 7), Reynolds number based on the lid velocity (Re = 100, 150, and 200), and amplitude of the sinusoidal function (A = 0.25, 0.5, and 1), keeping the solid-fluid thermal conductivity ratio fixed as K = 5. The flow and thermal fields are analyzed through streamline and isotherm plots for various Ha, Ri, Re, and Pr. Furthermore, the drag coefficient on the moving lid and Nusselt numbers on heated surfaces are also computed to understand the effects of Ha, Ri, Re, Pr, and n on them. It is observed that the drag on the moving lid decreases with Re and increases with Ri and Ha, however, remains insensitive with Pr. The heat-transfer rate from the hot right wall increases as usual with Re, Pr, and Ri but decreases with Ha. The sinusoidally heated bottom wall shows a decrease in the heat-transfer rate with increasing Pr, and at higher Pr, it also decreases with Re. Furthermore, increasing magnetic field strength causes an increase in the heat-transfer rate from the bottom wall. It also decreases with decreasing value of the amplitude of the sinusoidal function.