In medical image segmentation tasks, the domain gap caused by the difference in data collection between training and testing data seriously hinders the deployment of pre-trained models in clinical practice. Continual Test-Time Adaptation (CTTA) aims to enable pre-trained models to adapt to continuously changing unlabeled domains, providing an effective approach to solving this problem. However, existing CTTA methods often rely on unreliable supervisory signals, igniting a self-reinforcing cycle of error accumulation that culminates in catastrophic performance degradation. To overcome these challenges, we propose a CTTA via Semantic-Prompt-Enhanced Graph Clustering (SPEGC) for medical image segmentation. First, we design a semantic prompt feature enhancement mechanism that utilizes decoupled commonality and heterogeneity prompt pools to inject global contextual information into local features, alleviating their susceptibility to noise interference under domain shift. Second, based on these enhanced features, we design a differentiable graph clustering solver. This solver reframes global edge sparsification as an optimal transport problem, allowing it to distill a raw similarity matrix into a refined and high-order structural representation in an end-to-end manner. Finally, this robust structural representation is used to guide model adaptation, ensuring predictions are consistent at a cluster-level and dynamically adjusting decision boundaries. Extensive experiments demonstrate that SPEGC outperforms other state-of-the-art CTTA methods on two medical image segmentation benchmarks. The source code is available at https://github.com/Jwei-Z/SPEGC-for-MIS.