  We study the numerical approximation of the stochastic heat equation with a
distributional reaction term. Under a condition on the Besov regularity of the
reaction term, it was proven recently that a strong solution exists and is
unique in the pathwise sense, in a class of H\"older continuous processes. For
a suitable choice of sequence $(b^k)_{k\in \mathbb{N}}$ approximating $b$, we
prove that the error between the solution $u$ of the SPDE with reaction term
$b$ and its tamed Euler finite-difference scheme with mollified drift $b^k$,
converges to $0$ in $L^m(\Omega)$ with a rate that depends on the Besov
regularity of $b$. In particular, one can consider two interesting cases:
first, even when $b$ is only a (finite) measure, a rate of convergence is
obtained. On the other hand, when $b$ is a bounded measurable function, the
(almost) optimal rate of convergence $(\frac{1}{2}-\varepsilon)$-in space and
$(\frac{1}{4}-\varepsilon)$-in time is achieved. Stochastic sewing techniques
are used in the proofs, in particular to deduce new regularising properties of
the discrete Ornstein-Uhlenbeck process.
