  We study stochastic reaction--diffusion equation $$ \partial_tu_t(x)=\frac12
\partial^2_{xx}u_t(x)+b(u_t(x))+\dot{W}_{t}(x), \quad t>0,\, x\in D $$ where
$b$ is a generalized function in the Besov space
$\mathcal{B}^\beta_{q,\infty}({\mathbb R})$, $D\subset{\mathbb R}$ and $\dot W$
is a space-time white noise on ${\mathbb R}_+\times D$. We introduce a notion
of a solution to this equation and obtain existence and uniqueness of a strong
solution whenever $\beta-1/q\ge-1$, $\beta>-1$ and $q\in[1,\infty]$. This class
includes equations with $b$ being measures, in particular, $b=\delta_0$ which
corresponds to the skewed stochastic heat equation. For $\beta-1/q > -3/2$, 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
generalizes the results of Gy\"ongy 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 L\^e~(2020).
