Hyperspectral images (HSIs) are often degraded by complex mixed noise during acquisition and transmission, making effective denoising essential for subsequent analysis. Recent hybrid approaches that bridge model-driven and data-driven paradigms have shown great promise. However, most of these approaches lack effective alternation between different priors or modules, resulting in loosely coupled regularization and insufficient exploitation of their complementary strengths. Inspired by tensor robust principal component analysis (TRPCA), we propose a novel deep unfolding network (DU-TRPCA) that enforces stage-wise alternation between two tightly integrated modules: low-rank and sparse. The low-rank module employs thresholded tensor singular value decomposition (t-SVD), providing a widely adopted convex surrogate for tensor low-rankness and has been demonstrated to effectively capture the global spatial-spectral structure of HSIs. The Top-K sparse transformer module adaptively imposes sparse constraints, directly matching the sparse regularization in TRPCA and enabling effective removal of localized outliers and complex noise. This tightly coupled architecture preserves the stage-wise alternation between low-rank approximation and sparse refinement inherent in TRPCA, while enhancing representational capacity through attention mechanisms. Extensive experiments on synthetic and real-world HSIs demonstrate that DU-TRPCA surpasses state-of-the-art methods under severe mixed noise, while offering interpretability benefits and stable denoising dynamics inspired by iterative optimization. Code is available at https://github.com/liangli97/TRPCA-Deep-Unfolding-HSI-Denoising.