A highly accurate numerical method is proposed and analyzed for nonlinear strongly degenerate elliptic equation which has low regularity solution. In \(L^{1}\) space, we prove the existence and uniqueness of solution. The key and difficulty of the proposed method is to obtain Puiseux series expansion of solution about degenerate point. With integral expression of solution, by iteration method we recovery the Puiseux series expansion which contains an undetermined parameter, named as augmented variable. Since the degeneracy can be depicted by the Puiseux series on the singular domain, degenerate singular problem can be solved by coupling a uniform grid on a regular region with high order numerical scheme. The main advantage of the proposed method is that the convergence order of a global bad problem is determined by the convergence order of a good problem on the subdomain. The numerical solution of the augmented variable related to singularity is obtained from the high order scheme on the regular region, so the effect of singularity on the calculation region is essentially reduced. Although discrete scheme involves augmented variables leading to great difficulty to error estimate, using the properties of the Puiseux series expansion that we have recovered, a rigorous error estimate is conducted and fourth order accuracy is obtained. The accuracy of the proposed method is validated through a few numerical examples.