Single-cell RNA sequencing (scRNA-seq) offers significant opportunities to reveal cellular heterogeneity and diversity. Accurate cell type identification is critical for downstream analyses and understanding the mechanisms of heterogeneity. However, challenges arise from the high dimensionality, sparsity, and noise of scRNA-seq data. While various low-rank representation (LRR)-based clustering methods have been developed, many existing approaches may inaccurately capture relationships or conflate true patterns with noise. To address these limitations, we introduce a novel clustering algorithm that integrates low-rank matrix decomposition with local graph regularization (LRMGC). This approach applies a tri-decomposition strategy to the representation matrix to derive an aligned core matrix, and characterizes the "distance" between cells in a lower-dimensional space through a local manifold regularization term. Rather than relying on the kernel norm of the representation matrix, the Schatten p-norm is applied to the core matrix to robustly learn the similarity matrix against noise and outliers, while maintaining the high-dimensional noisy data's underlying subspace structure for accurate and robust clustering. Additionally, the final similarity matrix is obtained by applying the angular alignment strategy on the similarity matrix. Comprehensive experiments and comparisons with advanced methods on scRNA-seq datasets demonstrate LRMGC's superior performance and reliability in uncovering cell type composition. Furthermore, a variety of downstream analyses, such as marker gene identification, functional enrichment analysis, rare cell recognition, and cell-cell communication, also demonstrate the effectiveness of LRMGC.