Regularity for a strongly degenerate equation with explicit $u$-dependence

We consider local weak solutions of widely degenerate elliptic PDEs of the type \begin{equation} \label{equazione mia} $\mathrm{d}\mathrm{i}\mathrm{v}$\Biggl(a(x)(|Du|-1)^{p-1}_+\frac{Du}{|Du|}\Biggr)=b(x,u) \ \ $\text{ in }$Ω, \end{equation} where $2\le p<\mathrm{\infty },\mathrm{\Omega }$ is an open subset of ${\mathbb{R}}^n,n>2,$ and $(\cdot {)}_{+}$ stands for the positive part. We establish a higher differentiability result for the composition of the gradient with a suitable function that vanishes in the unit ball for the gradient, under suitable assumptions on the datum $b(x,u)$ and the coefficient $a(x)\mathrm{.}$ The novelty here with respect to previous papers on the subject is that the right hand side explicitly depends on the solution $u\mathrm{.}$