Aerial robotics for transporting suspended payloads as the form of freely-floating manipulator are growing great interest in recent years. However, the force/torque caused by payload and residual dynamics will introduce unmodeled perturbations to the aerial robotics, which negatively affects the closed-loop performance. Different from estimation-like methods, this paper proposes Neural Predictor, a learning-based approach to model force/torque induced by payload and residual dynamics as a dynamical system. It yields a hybrid model that combines the first-principles dynamics with the learned dynamics. The hybrid model is then integrated into a MPC framework to improve closed-loop performance. Effectiveness of proposed framework is verified extensively in both numerical simulations and real-world flight experiments. The results indicate that our approach can capture force/torque caused by suspended payload and residual dynamics accurately, respond quickly to the changes of them and improve the closed-loop performance significantly. In particular, Neural Predictor outperforms a state-of-the-art learning-based estimator and has reduced the force and torque estimation errors by up to 66.15% and 33.33% while requiring less samples. The code of proposed Neural Predictor can be found at https://github.com/NPU-RCIR/Neural-Predictor.git.