In addition to Gaussian noise, there is sparse noise with impulsive properties in the signal acquisition process. The common robust sparse signal recovery models can recover the original sparse signal under sparse noise environment. However, in many practical applications, the structural sparsity of the original signal, for example, gradient sparsity needs to be considered. In order to recover the sparse structure of the original high-dimensional signal from the coexistence of sparse noise and Gaussian noise, this paper proposed two nonconvex and nonsmooth optimization models based on truncated L₁-L₂ total variation (TV) and 3D truncated L₁-L₂ TV, respectively. These optimization models were solved by the proximal alternating linearized minimization algorithm with extrapolation, and the sub-problems involved were solved by the proximal convex difference algorithm with extrapolation. Under the assumption that the potential function has Kurdyka-Lojasiewicz (KL) property, the convergence analysis of these algorithms was given. The numerical experiments test grey images with Gaussian noise, color images with mixed noise, grey video with mixed noise and so on. The peak signal-to-noise ratio (PSNR) was used as the evaluation criterion for recovered quality. The experimental results show that the new models can correctly recover the original structured sparse signal, and have better PSNR values in the same noisy environment.