Pulmonary nodules and masses are crucial imaging features in lung cancer screening that require careful management in clinical diagnosis. Despite the success of deep learning-based medical image segmentation, the robust performance on various sizes of lesions of nodule and mass is still challenging. In this paper, we propose a multi-scale neural network with scale-aware test-time adaptation to address this challenge. Specifically, we introduce an adaptive Scale-aware Test-time Click Adaptation method based on effortlessly obtainable lesion clicks as test-time cues to enhance segmentation performance, particularly for large lesions. The proposed method can be seamlessly integrated into existing networks. Extensive experiments on both open-source and in-house datasets consistently demonstrate the effectiveness of the proposed method over some CNN and Transformer-based segmentation methods. Our code is available at https://github.com/SplinterLi/SaTTCA.