This study aims to introduce a new strategy for anticipating the behaviour of human-driven vehicles (HDVs) and designing trajectories for connected and automated vehicles (CAVs) at signalized intersections under mixed traffic scenarios. To tackle the challenge of unreliable HDV trajectory predictions stemming from driving unpredictability, a recursive framework is developed. This framework integrates real-time tracking data from both traffic detectors and CAVs, continuously updating HDV predictions. The proposed approach employs the updated predictions to formulate optimal control problems recursively to optimize or adjust CAV trajectories, enhancing travel and energy efficiency. Besides, the recomputing of CAV trajectories will only be conducted when the variation in predictions rises to a certain threshold, balancing efficiency and computing consumption, inspired and modified based on MPC methods. The application of the Pontryagin maximum principle aids in finding solutions efficiently by transforming necessary conditions into a system of equations and consolidating elementary unconstrained and constrained arcs. Numerical simulations were carried out to evaluate the performance of the proposed recursive framework, revealing its superiority over the one-time approach, particularly in isolated intersections with high traffic demands. Additionally, the recursive framework exhibited more robust and effective enhancements throughout the road network.