We prove a sharp higher differentiability result for local minimizers of functionals of the form $$\mathcal{F}\left(w,Ω\right)=\int_Ω\left[ F\left(x,Dw(x)\right)-f(x)\cdot w(x)\right]dx$$ with non-autonomous integrand $F(x,ξ)$ which is convex with respect to the gradient variable, under $p$-growth conditions, with $1<p<2$. The main novelty here is that the results are obtained assuming that the partial map $x\mapsto D_ξF(x,ξ)$ has weak derivatives in some Lebesgue space $L^q$ and the datum $f$ is assumed to belong to a suitable Lebesgue space $L^r$. We also prove that it is possible to weaken the assumption on the datum $f$ and on the map $x\mapsto D_ξF(x,ξ)$, if the minimizers are assumed to be a priori bounded.