Robust motion planning under uncertainty is critical for unlocking real-world robotics applications. This paper introduces SupeR-MPC, a computationally-efficient, sensitivity-aware, chance-constrained optimization framework that systematically accounts for multiple sources of uncertainty, including state estimation error, model parameter uncertainty, obstacle localization error, and process noise. This approach advances sensitivityaware robust control by integrating chance-constrained optimization to handle the uncertainty models of Kalman-filtering methods. To demonstrate robustness against multiple uncertainty sources, SupeR-MPC was validated on a range of systems and environments, from a simple 2D example to a multi-agent dynamic obstacle avoidance scenario. Comparisons against existing MPC methods show that SupeR-MPC significantly improves constraint satisfaction and robustness while maintaining real-time computational efficiency. These results highlight the effectiveness of sensitivity-aware chance constraints in enhancing real-world robotic decision making under uncertainty.