This contribution aims at presenting and generalizing a recent work of Hernández, Jara and Valentim [10]. We consider the weakly asymmetric version of the so-called discrete Atlas model, which has been introduced in [10]. Precisely, we look at some equilibrium fluctuation field of a weakly asymmetric zero-range process which evolves on a discrete half-line, with a source of particles at the origin. We prove that its macroscopic evolution is governed by a stochastic heat equation with Neumann or Robin boundary conditions, depending on the range of the parameters of the model.