We establish weak existence and uniqueness for random field solutions of the one-dimensional SPDE \[ d_tX_t = \frac{1}{2}ΔX_t +h(X_t)+ \sqrt{X_t}\dot{W}, \quad t\geq 0,\] where $\dot{W}$ is space-time white noise and $h$ is a bounded drift with $h(0)\geq 0$. The proof relies on an extension of the duality relation of the super-Brownian motion, which allows us to treat a broad class of admissible drifts, including functions that are non-Lipschitz or discontinuous at zero. In particular, well-posedness is derived for certain drifts that are Hölder continuous at zero with exponent $α\in(0,1)$. We also allow discontinuous drifts of the form $h(x) = b_0\unicode{x1D7D9}_{x = 0} + b_1\unicode{x1D7D9}_{x>0},$ where $b_0 \geq 0$, $b_1 \in \mathbb{R}$. Additionally, if $h(0)=0$ and the initial condition is continuous and compactly supported, we show that the Lebesgue measure of the non-zero set of $X$ is finite. The proofs are based on duality. We use a log-Laplace equation, which is perturbed by jump noise as the equation for the dual process, and the jumps of the dual process are allowed to take infinite values. We believe that the results for the dual process are also of independent interest. Under suitable assumptions on $h$ we also prove survival of $X$ with positive probability, using rescaling and comparison to the KPP-equation with branching noise.