  In this paper we prove an effective equidistribution result for both
primitive and non-primitive points on certain expanding horocycle sections in
ASL$(2,\mathbb{Z}) \backslash $ASL$(2,\mathbb{R})$. This provides an effective
version of a result recently proven by the author and generalises recent work
by Einsiedler, Luethi and Shah who prove similar results in the context of the
unit tangent bundle of the modular surface.
