  We establish some higher differentiability results for solution to
non-autonomous obstacle problems of the form
  \begin{equation*}
  \min \left\{\int_{\Omega}f\left(x, Dv(x)\right)dx\,:\, v\in
  \mathcal{K}_\psi(\Omega)\right\},
  \end{equation*} where the function $f$ satisfies $p-$growth conditions with
respect to the gradient variable, for $1<p<2$, and $\mathcal{K}_\psi(\Omega)$
is the class of admissible functions. Here we show that, if the obstacle $\psi$
is bounded, then a Sobolev regularity assumption on the gradient of the
obstacle $\psi$ transfers to the gradient of the solution, provided the partial
map $x\mapsto D_\xi f(x,\xi)$ belongs to a Sobolev space, $W^{1, p+2}$. The
novelty here is that we deal with subquadratic growth conditions with respect
to the gradient variable, i.e. $f(x, \xi)\approx a(x)|\xi|^p$ with $1<p<2,$ and
where the map $a$ belongs to a Sobolev space.
