This paper proposes Super-Twisting sliding mode controllers (STSMCs) to solve the trajectory tracking problem of a cargo Uncrewed Aerial Vehicle (UAV), while actively reducing the oscillations of the cable-suspended payload hanging from the quadrotor. In order to damp the oscillatory behavior of the cargo, we propose sliding surfaces which include the dynamics of the load swing angles. The proof of stability is carried out through Lyapunov analysis, resulting in finite-time convergence of the sliding surfaces and their derivatives, and asymptotic convergence of the tracking errors. Simulations in Matlab and Simulink are presented to show the effectiveness of the designed STSMCs, considering a small-scale cargo drone which performs a circular and a helical trajectory. Under the proposed controllers, the UAV is able to track the references, while the load swing angles do not exceed the magnitude of 2.5 ◦ . Our approach allows to reduce load oscillations by more than 50% when compared to STSMCs without active swing reduction.