This paper has presented an iterative approach to synthesizing hybrid polynomial-DNN controllers for nonlinear systems such that the closed-loop system can be both well-performing and easily verified upon required properties. Our approach has creatively integrated low-degree polynomial fitting and knowledge distillation into the RL method during the construction process. Thanks to the special feature of the hybrid controller, the controlled system can be transformed into the polynomial form. By utilizing the SOS relaxation method and solving LMI constraints efficiently, we can generate barrier certificates to verify the safety properties of the nonlinear control systems equipped with our synthesized hybrid controllers. Extensive experiments consistently demonstrate the effectiveness and scalability of the proposed approach.