In this paper we develop and analyze two energy-preserving hybrid asymptotic augmented finite volume methods on uniform grids for nonlinear weakly degenerate and strongly degenerate wave equations. In order to deal with the degeneracy, we introduce an intermediate point to divide the whole domain into singular subdomain and regular subdomain. Then Puiseux series asymptotic technique is used in singular subdomain and augmented finite volume scheme is used in regular subdomain. The keys of the method are the recovery of Puiseux series in singular subdomain and the appropriate combination of singular and regular subdomain by means of augmented variables associated with the singularity. Although the effect of singularity on the calculation domain is conquered by the Puiseux series reconstruction technique, it also brings difficulties to the theoretical analysis. Based on the idea of staggered grid, we overcome the difficulties arising from the augmented variables related to singularity for the construction of conservation scheme. The discrete energy conservation and convergence of the two energy-preserving methods are demonstrated successfully. The advantages of the proposed methods are the energy conservation and the global convergence order determined by the regular subdomain scheme. Numerical examples on weakly degenerate and strongly degenerate under different nonlinear functions are provided to demonstrate the validity and conservation of the proposed method. Specially, the conservation of discrete energy is also ensured by using the proposed methods for both the generalized Sine-Gordeon equation and the coefficient blow-up problem.