<p>A novel hybrid augmented approach is proposed for solving nonlinear degenerate elliptic equations. A robust technique is constructed to recover the Puiseux series expansion, which incorporates augmented variables, specifically for the complete nonlinear degenerate coefficient of the elliptic operator. The method provides a more general framework for enhancing the accuracy of solutions to general nonlinear degenerate problems. Since the augmented variables are organically connected to singularities and bridge the singular subdomain with the regular subdomain, they can be approximated using higher-order schemes on uniform grids within the regular subdomain. In this paper, the degenerate problems are discretized by finite volume method. An outstanding advantage of the proposed method is that the convergence order of the solution on the entire domain is determined by the convergence properties of the scheme applied to the well-posed problem on the regular subdomain. A rigorous error estimation for the hybrid augmented finite volume method (AFVM) is established by the energy method. The validity and effectiveness of the proposed method are demonstrated through a comprehensive set of physically motivated nonlinear degenerate examples, including chemical reactor problems with coefficient blowing up, stationary heat conduction degenerate problems, porous medium equations (PMEs) with or without mass sources, heat-conduction non-degenerate problems based on the Boltzmann fourth-power law, two-dimensional (2D) nonlinear elliptic equation and strong nonlinear degenerate problems. It is shown that both degenerate and non-degenerate cases achieve the same level of accuracy as predicted by our theoretical findings.</p>