Dynamic electric vehicle routing problem (DEVRP) is a variation of Electric vehicle routing problem (EVRP), which incorporates real-time changes in customer requirements and road conditions during route planning, brings the problem closer to real-world scenarios while introducing additional complexities and challenges. In the literature, existing approaches are typically categorized into three types: exact methods, metaheuristic methods, and machine learning methods. Among these approaches, both exact and heuristic methods tend to exhibit slow response time when dealing with complex and dynamic problems, each having its specific limitations. Moreover, machine learning methods have been limited in application in the field of path planning, and existing models struggle to effectively address the complexities of DEVRP. To address these challenges, we propose a heuristic algorithm with enhanced graph transformer networks (GTN-HA), which consists of three key phases, namely the generation of initial solutions, heuristic search and dynamic response. Specifically, the initial solutions is generated by the proposed enhanced GTN, which incorporates time windows and constraints in the endcoder and decoder process. Moreover, the REINFORCE with a greedy rollout baseline is employed to update the parameters of the proposed enhanced GTN. In the heuristic search phase, the generated initial solutions serves as the initial population to obtain the optimal solutions. At the stage of dynamic response, the optimal solutions are utilized to update the enhanced GTN model. To verify the performance of the proposed method, we conducted comprehensive empirical experiments using various heuristic algorithms on DEVRP instances with diverse characteristic.