Summary: To restore images with clear edge details and no staircase artifacts from degraded versions, this paper incorporates the \(\ell_2\) plus \(\ell_0\) data fidelity and non-convex high-order total variation regularizer to establish an optimization model for eliminating mixed Gaussian-impulse noise. Among them, the \(\ell_2\) fidelity is adopted to suppress Gaussian noise, while the \(\ell_0\)-norm is more suitable for detecting and removing impulse noise. In addition, the non-convex regularization displays excellent performance in overcoming the staircase effect and preserving edge details. Computationally, by using the iteratively reweighted \(\ell_1\) algorithm and variable splitting method, this work designs a modified alternating minimization method to solve the optimization problem we construct. In theory, the convergence proof of our resulting algorithm is also presented. Finally, in the experimental section, we conduct extensive numerical experiments on degraded images and compare with other existing techniques. From the intuitive effects and restoration accuracy, it follows that our newly proposed method is effective and competitive for image deblurring and mixed noise removal.