Multi-view clustering has been empirically shown to improve learning performance by leveraging the inherent complementary information across multiple views of data. However, in real-world scenarios, collecting strictly aligned views is challenging, and learning from both aligned and unaligned data becomes a more practical solution. Partially View-aligned Clustering aims to learn correspondences between misaligned view samples to better exploit the potential consistency and complementarity across views, including both aligned and unaligned data. However, most existing PVC methods fail to leverage unaligned data to capture the shared semantics among samples from the same cluster. Moreover, the inherent heterogeneity of multi-view data induces distributional shifts in representations, leading to inaccuracies in establishing meaningful correspondences between cross-view latent features and, consequently, impairing learning effectiveness. To address these challenges, we propose a Semantic MAtching contRasTive learning model (SMART) for PVC. The main idea of our approach is to alleviate the influence of cross-view distributional shifts, thereby facilitating semantic matching contrastive learning to fully exploit semantic relationships in both aligned and unaligned data. Extensive experiments on eight benchmark datasets demonstrate that our method consistently outperforms existing approaches on the PVC problem.