We consider local weak solutions to PDEs of the type \( -\,\textrm{div}\left( (\vert Du\vert -\lambda )_{+}^{p-1}\frac{Du}{\vert Du\vert }\right) =f\,\,\,\,\,\,\,\text {in}\,\,\Omega , \) where \(1<p<\infty \) , \(\Omega \) is an open subset of \(\mathbb {R}^{n}\) for \(n\ge 2\) , \(\lambda \) is a positive constant and \((\,\cdot \,)_{+}\) stands for the positive part. Equations of this form are widely degenerate for \(p\ge 2\) and widely singular for \(1<p<2\) . We establish higher differentiability results for a suitable nonlinear function of the gradient Du of the local weak solutions, assuming that f belongs to the local Besov space \(B^{(p-2)/p}_{p',1,loc}(\Omega )\) when \(p>2\) , and that \(f\in L_{loc}^{{\frac{np}{n(p-1)+2-p}}}(\Omega )\) if \(1<p\le 2\) . The conditions on the datum f are essentially sharp. As a consequence, we obtain the local higher integrability of Du under the same minimal assumptions on f. For \(\lambda =0\) , our results give back those contained in Clop et al. (Bull Math Sci 13(12):2350008, 2023) and Irving and Koch (Adv Nonlinear Anal 12(1):20230110, 2023).
