Hyperspectral images (HSIs) are inevitably contaminated by the mixed noise (such as Gaussian noise, impulse noise, deadlines, and stripes), which could influence the subsequent processing accuracy. Generally, HSI restoration can be transformed into the low-rank matrix recovery (LRMR). In the LRMR, the nuclear norm is widely used to substitute the matrix rank, but its effectiveness is still worth improving. Besides, the l0 -norm cannot capture the sparse noise's structured sparsity property. To handle these issues, the adaptive rank and structured sparsity corrections (ARSSC) are presented for HSI restoration. The ARSSC introduces two convex regularizers, that is: 1) the rank correction (RC) and 2) the structured sparsity correction (SSC), to, respectively, approximate the matrix rank and the l2,0 -norm. The RC and the SSC can adaptively offset the penalization of large entries from the nuclear norm and the l2,1 -norm, respectively, where the larger the entry, the greater its offset. Therefore, the proposed ARSSC achieves a tighter approximation of the noise-free HSI low-rank structure and promotes the structured sparsity of sparse noise. An efficient alternative direction method of multipliers (ADMM) algorithm is applied to solve the resulting convex optimization problem. The superiority of the ARSSC in terms of the mixed noise removal and spatial-spectral structure information preserving, is demonstrated by several experimental results both on simulated and real datasets, compared with other state-of-the-art HSI restoration approaches.