This study investigates the effects of Biot heat flux and injection/suction on magnetohydrodynamic natural convection flow over a cylindrical structure, relevant to applications like copper coating, chip coating, crude oil refinement, and paper production. The fluid rheology is modeled using the non-Newtonian Carreau fluid model. Similarity variables simplify the governing equations, which are solved using the homotopy analysis method. A comprehensive parametric analysis is conducted to investigate the effects of convective heat flux, and suction/injection on heat transfer characteristics and fluid flow dynamics. The findings indicate that Biot numbers enhance temperature distribution and increase heat transfer rates while improving skin friction. Conversely, the suction parameter decreases temperature and velocity profiles, whereas injection improves both. These results provide valuable insights for developing efficient and sustainable industrial processes.