Recovering a drone on a disturbed water surface remains a significant challenge in maritime robotics. In this paper, we propose a unified framework for robot-assisted drone recovery on a wavy surface that addresses two major tasks: Firstly, accurate prediction of a moving drone's position under wave-induced disturbances using KalmanNet Plus Plus (KalmanNet++), a Neural Network Aided Kalman Filtering we proposed. Secondly, effective motion planning using the desired position we got for a manipulator via Receding Horizon Model Predictive Control (RHMPC). Specifically, we compared multiple prediction methods and proposed KalmanNet Plus Plus to predict the position of the UAV, thereby obtaining the desired position. The KalmanNet++ predicts the drone's future position 0.1\,s ahead, while the manipulator plans a capture trajectory in real time, thus overcoming not only wave-induced base motions but also limited constraints such as torque constraints and joint constraints. For the system design, we provide a collaborative system, comprising a manipulator subsystem and a UAV subsystem, enables drone lifting and drone recovery. Simulation and real-world experiments using wave-disturbed motion data demonstrate that our approach achieves a high success rate - above 95\% and outperforms conventional baseline methods by up to 10\% in efficiency and 20\% in precision. The results underscore the feasibility and robustness of our system, which achieves state-of-the-art performance and offers a practical solution for maritime drone operations.

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