  Consider the $[0,1]$-valued continuous random field solution $(u_t(x))_{t\geq
0, x\in \mathbb R}$ to the one-dimensional stochastic heat equation \[
\partial_t u_t = \frac{1}{2}\Delta u_t + b(u_t) + \sqrt{u_t(1-u_t)} \dot W, \]
where $b(1)\leq 0\leq b(0)$ and $\dot W$ is space-time white noise. In this
paper, we establish the weak existence and uniqueness of the above equation for
a class of drifts $b(u)$ that may be irregular at the points where the noise
coefficient is non-Lipschitz and degenerate, specifically at $u=0$ or $u=1$.
This class of drifts includes non-Lipschitz drifts like $b(u) = u^q(1-u)$ for
every $q\in (0,1)$, and some discontinuous drifts like $b(u) = \mathbf
1_{(0,1]}(u)-u$. This demonstrates a regularization effect of the
multiplicative space-time white noise without the standard assumption that the
noise coefficient is Lipschitz and non-degenerate.
  The method we apply is a further development of a moment duality technique
that uses branching-coalescing Brownian motions as the dual particle system. To
handle an irregular drift in the above equation, particles in the dual system
are allowed to have a number of offspring with infinite expectation, and even
an infinite number of offspring with positive probability. We show that, even
though the branching mechanism with an infinite number of offspring causes
explosions in finite time, immediately after each explosion, the total
population comes down from infinity due to the coalescing mechanism. Our
results on this dual particle system are of independent interest.
