Second-order regularity for weighted widely degenerate problems with explicit $u$-dependence

We consider local weak solutions of widely degenerate elliptic PDEs of the type \begin{equation} \label{equazione mia} \mathrm{div}\Biggl(|x|^β(|Du|-1)^{p-1}_+\frac{Du}{|Du|}\Biggr)=\frac{|u|^{q-2}u}{|x|^α} \ \ \text{ in }Ω, \end{equation} where $2\leq p0$ are fixed exponents, $Ω$ is