In this work we are concerned with an energy functional related to the Thomas-Fermi-Dirac-von Weizsäcker (TFDW) model, a physical model describing ground state electron configurations of many-body systems. The model we consider comes from replacing the Newtonian potential in TFDW by an external potential of longer range. It has been recently shown that, unlike in the TFDW setting and the case the charge is zero, the slow decay of the external potential makes it attractive enough as for allowing for existence of ground states for all $M>0$ and $Z>0$. Our main result describes the structure of minimizers in the limit $Z\to0$ by means of an asymptotic expansion of the energy.