In this paper, a neural-adaptive, prescribed-performance trajectory controller is proposed to stabilize a flexible cable-towed aerial-recovery drogue subjected to actuator constraints, unmeasurable cable tensions, and airflow disturbances. The towed drogue’s six-degree-of-freedom (6-DOF) dynamics are formulated in a nominal affine nonlinear form based on the flexibly cable-drogue system’s dynamics. To accurately reconstitute and compensate for unmeasurable lumped disturbances, including the effects of unmeasurable tensions and unstable airflows, an estimator-based minimal learning parameter neural network (EMLPNN) is established for each subsystem of the drogue dynamics. Then, an EMLPNN-based prescribed performance controller is proposed to stabilize the aerial-recovery-drogue trajectory with prescribed performance. Moreover, the problem of the actuator constraints is handled by establishing an auxiliary system whose states are employed to compensate for the angular-rate control law. The closed-loop stability is analyzed. Drogue stabilization simulations under airflow disturbances are conducted for verification of the control performance and effectiveness.