Accurate lesion segmentation in medical imaging is critical for medical diagnosis and treatment. Lesions' diverse and heterogeneous characteristics often present a distinct long-tail distribution, posing difficulties for automatic methods. Currently, interactive segmentation approaches have shown promise in improving accuracy, but still struggle to deal with tail features. This triggers a demand of effective utilizing strategies of user interaction. To this end, we propose a novel point-based interactive segmentation model called Clustering-Aided Interactive Segmentation Network (CAISeg) in 3D medical imaging. A customized Interaction-Guided Module (IGM) adopts the concept of clustering to capture features that are semantically similar to interaction points. These clustered features are then mapped to the head regions of the prompted category to facilitate more precise classification. Meanwhile, we put forward a Focus Guided Loss function to grant the network an inductive bias towards user interaction through assigning higher weights to voxels closer to the prompted points, thereby improving the responsiveness efficiency to user guidance. Evaluation across brain tumor, colon cancer, lung cancer, and pancreas cancer segmentation tasks show CAISeg's superiority over the state-of-the-art methods. It outperforms the fully automated segmentation models in accuracy, and achieves results comparable to or better than those of the leading point-based interactive methods while requiring fewer prompt points. Furthermore, we discover that CAISeg possesses good interpretability at various stages, which endows CAISeg with potential clinical application value.