Public transportation has been identified as a viable solution to mitigate traffic congestion. Transit signal priority (TSP) control, which is widely used at signalized intersections, has been recognized as a practical strategy to improve the efficiency and reliability of bus operations. However, traditional TSP control may fall short of efficiency and is facing several challenges of negative externalities for non-transit users and the need to handle conflicting priority requests. Recent studies have proposed the use of reinforcement learning (RL) methods to identify efficient traffic signal control (TSC). Some of these studies on RL-based TSC have incorporated the concept of max-pressure (MP), which is a maximal weight-matching algorithm to minimize queue sizes. Nevertheless, the existing RL-based TSC methods focus on private vehicles and cannot adequately distinguish between buses and private vehicles. In prior research, RL-based control has been implemented within the context of bus rapid transit (BRT) systems. This study proposes a novel RL-based TSC strategy that leverages the MP concept and extends it to incorporate TSP control. This is the first implementation of RL-based TSP control within the mixed-traffic road network. A significant innovation of this research is the introduction of the priority factor (PF), which is designed to prioritize bus movements at signalized intersections. The proposed RL-based TSP with PF control seeks to balance the competing objectives of enhancing bus operations while mitigating adverse impacts on non-transit users. To evaluate the performance of the proposed TSP method with the PF mechanism, simulations were conducted on an arterial and a grid network under dynamic traffic conditions. The simulation results demonstrated that the proposed TSP with PF not only reduces bus travel times and resolves conflicts between priority requests but also does not make a significant negative impact on passenger car operations. Furthermore, the PF can be dynamically assigned according to the number of passengers on each bus, suggesting the potential for the proposed approach to be applied in various traffic management scenarios.