Contrastive graph clustering methods have attracted increasing attention due to their ability to jointly capture node semantics and structural information in an unsupervised manner. However, most existing methods rely heavily on shallow adjacency structures or sample pairs constructed in the feature space, without explicitly modeling or enhancing the underlying graph structure. This leads to limited capability in preserving topological patterns and thus restricts clustering performance. Moreover, conventional contrastive objectives struggle to capture intra-cluster structural patterns and inter-cluster differences. To address these limitations, we propose CGCSE, a contrastive graph clustering framework with structure-aware enhancement. CGCSE introduces a structural enhancement module to explicitly model graph semantics and construct a structure-aware similarity graph, generates structure-perturbed views guided by spectral features, and employs a clustering-aware contrastive loss to jointly optimize node representations and clustering discriminability. Extensive experiments on five public datasets demonstrate that CGCSE consistently outperforms existing methods in clustering performance.