Federated learning (FL) enables model training on mobile clients (M Cs) while protecting data privacy by keeping the data local. However, the data and system heterogeneity among MCs can significantly undermine model performance, slow convergence, and increase energy consumption. To achieve green and efficient edge intelligence, we propose an energy consumption optimization problem under the FL framework for mobile edge computing networks in this paper. Our goal is to reduce the energy consumption of MCs and improve the FL model performance. Then we design a heterogeneity-aware client selection and gradient optimization (HCSGO) algorithm. Specifically, HCSGO selects MCs based on data, computation, and communication quality to mitigate the impact of heterogene-ity on model performance, while leveraging a residual gradient mechanism to optimize gradient aggregation and accelerate convergence. Experiment results demonstrate that the proposed algorithm achieves the lowest energy consumption and improves the model performance compared to the baselines.