Microservice deployment in cloud computing is a challenging combinatorial optimization problem due to the complex dependencies among microservices and the intricate trade-offs among different QoS requirements, e.g., minimizing Energy Consumption (EC) vs. minimizing Communication Overhead (CO). Recently, some hyper-heuristics methods, particularly Genetic Programming Hyper-Heuristics (GPHH), have been proposed to automatically generate heuristics for solving dynamic microservice deployment problems. Meanwhile, Large Language Models (LLMs) are becoming popular for solving various domain-specific problems thanks to their strong ability to learn problem-related knowledge. However, hybridizing GPHH with LLM by combining their abilities in solving complex optimization problems remains unexplored. In this paper, we propose an LLM-enhanced Genetic Programming Hyper-Heuristic (LLM-GPHH) algorithm to evolve heuristics for the dynamic deployment of applications composed of microservices, to jointly optimize EC and CO. Our experiments on real-world datasets demonstrate the effectiveness of the newly proposed LLM-GPHH.