We resort to approximation methods to produce a regularity theory for degenerate models, both in the variational and nonvariational settings. First, we consider a $p$-Poisson equation and approximate it with the Laplace operator. The method allows us to control the distance of the solutions to a harmonic function by a small parameter and a quantity of class $C^1$. This fact unlocks an $C^{1,1-}$ regularity result for the solutions to the $p$-Poisson problem. In the nonvariational context, we consider fully nonlinear equations degenerating as a power of the gradient. Here we detail how approximation methods characterize the optimal regularity of the solutions in terms of the degeneracy rate. A key step is a cancellation lemma in the viscosity sense.