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With the booming deployment of Internet of Things, health monitoring applications have gradually prospered. Within the recent COVID-19 pandemic situation, interest in permanent remote health monitoring solutions has raised, targeting to reduce contact and preserve the limited medical resources. Among the technological methods to realize efficient remote health monitoring, federated learning (FL) has drawn particular attention due to its robustness in preserving data privacy. However, FL can yield to high communication costs, due to frequent transmissions between the FL server and clients. To tackle this problem, we propose in this paper a communication-efficient federated learning (CEFL) framework that involves clients clustering and transfer learning. First, we propose to group clients through the calculation of similarity factors, based on the neural networks characteristics. Then, a representative client in each cluster is selected to be the leader of the cluster. Differently from the conventional FL, our method performs FL training only among the cluster leaders. Subsequently, transfer learning is adopted by the leader to update its cluster members with the trained FL model. Finally, each member fine-tunes the received model with its own data. To further reduce the communication costs, we opt for a partial-layer FL aggregation approach. This method suggests partially updating the neural network model rather than fully. Through experiments, we show that CEFL can save up to to 98.45% in communication costs while conceding less than 3% in accuracy loss, when compared to the conventional FL. Finally, CEFL demonstrates a high accuracy for clients with small or unbalanced datasets.