In this paper, an accurate and efficient augmented finite element method for nonlinear degenerate two point boundary value problem is presented. The main idea of the new method is to represent the solution by the Puiseux series expansion in a subinterval that contains the singularity of the solution. The end point of the subinterval is an augmented variable that needs to be determined. One significant advantage of the method is that the subinterval contains the singularity and the rest of domain is connected by the augmented variable naturally, so that a high-accuracy numerical scheme over the entire region can be obtained using a uniform mesh. The convergence order of the schemes for solving the degenerate problem is determined by the scheme on the subinterval in which the solution has no singularity. The new approach is robust for solving such degenerate problems. Numerical experiments are presented to verify the efficiency and accuracy of the new method.