We consider the following prototype problem: $$\begin{aligned} \left\{ \begin{aligned}&-\Delta u + M \frac{|\nabla u|^2}{u^\theta }=f&\hbox { in}\ \varOmega \\&u=0&\hbox { on}\ \partial \varOmega \end{aligned}\right. \end{aligned}$$ and we study the regularity of the gradient of a solution both in Morrey spaces and in fractional Sobolev spaces in correspondence of the regularity of the right-hand side.