Robust tensor recovery aims to reconstruct a multidimensional tensor from its observations contaminated by noise. In this brief, a new formulation based on correntropy and hybrid tensor sparsity measure is proposed for robust tensor recovery in the environment of impulsive noise. The robust correntropy measure has shown satisfactory robustness against large outliers in various scenarios recently. Meanwhile, in this formulation, the hybrid tensor sparsity measure combining the advantages of Tucker and CP tensor rank can better characterize the tensor sparsity. To solve the proposed formulation effectively, an efficient large-scale optimization algorithm is derived based on the framework of alternating direction method of multipliers (ADMM) and half-quadratic optimization technique. The results of multispectral image (MSI) data recovery indicate that the proposed algorithm can achieve robust tensor recovery in the environment of impulsive noise.