Aerial robotics for transporting suspended payloads as the form of freely-floating manipulator are growing great interest in recent years. However, the prior information of the payload, such as the mass, is always hard to obtain accurately in practice. The force/torque caused by payload and residual dynamics will introduce unmodeled perturbations to the system, 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 caused by payload and residual dynamics as a dynamical system. It results a hybrid model including both the first-principles dynamics and the learned dynamics. This 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 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 using less samples.