We investigate the macroscopic behavior of asymmetric attractive zero-range processes on $\mathbb{Z}$ where particles are destroyed at the origin at a rate of order $N^β$, where $β\in \mathbb{R}$ and $N\in\mathbb{N}$ is the scaling parameter. We prove that the hydrodynamic limit of this particle system is described by the unique entropy solution of a hyperbolic conservation law, supplemented by a boundary condition depending on the range of $β$. Namely, if $β\geqslant 0$, then the boundary condition prescribes the particle current through the origin, whereas if $β<0$, the destruction of particles at the origin has no macroscopic effect on the system and no boundary condition is imposed at the hydrodynamic limit.