We study stochastic reaction–diffusion equation ∂tut(x)=12∂xx2ut(x)+b(ut(x))+Ẇt(x),t>0,x∈D$$\begin{equation*} \partial _tu_t(x)=\frac{1}{2} \partial ^2_{xx}u_t(x)+b(u_t(x))+\dot{W}_{t}(x), \quad t>0,\, x\in {D} \end{equation*}$$where b$b$ is a generalized function in the Besov space Bq,∞β(R)$\mathcal {B}^\beta _{q,\infty }(\mathbb {R})$, D⊂R${D}\subset \mathbb {R}$ and Ẇ$\dot{W}$ is a space‐time white noise on R+×D$\mathbb {R}_+\times {D}$. We introduce a notion of a solution to this equation and obtain existence and uniqueness of a strong solution whenever β−1/q⩾−1$\beta -1/q\geqslant -1$, β>−1$\beta >-1$ and q∈[1,∞]$q\in [1,\infty ]$. This class includes equations with b$b$ being measures, in particular, b=δ0$b=\delta _0$ which corresponds to the skewed stochastic heat equation. For β−1/q>−3/2$\beta -1/q > -3/2$, we obtain existence of a weak solution. Our results extend the work of Bass and Chen (2001) to the framework of stochastic partial differential equations and generalize the results of Gyöngy and Pardoux (1993) to distributional drifts. To establish these results, we exploit the regularization effect of the white noise through a new strategy based on the stochastic sewing lemma introduced in Lê (2020). [ABSTRACT FROM AUTHOR]