This study introduces a non-convex fractional-order hyper-Laplacian variational model for Gaussian noise removal. It employs first the primal-dual hybrid gradient algorithm to solve problems involving overlapping group sparse structures. Additionally, this paper designs a new algorithm leveraging the framework of the Chambolle-Pock algorithm with convergence and aims to recover high-quality images. The model, integrating the overlapping group sparse structure of the hyper-Laplacian prior with the non-convex fractional-order total variation, exhibits superior performance in reducing the staircase effect and maintaining sharp edge contours. To further improve the performance of the algorithm, a semi-adaptive p(x) non-convex penalty weight assignment mechanism is designed by introducing the structure tensor, which according to the characteristics of each region of the image and the noise level. The effectiveness and superiority of the proposed algorithm in image denoising with simulation experiments and comparative analyses are fully verified.