  In this paper, we consider the notion of Cotton soliton within the framework
of almost Kenmotsu 3-$h$-manifolds. First we consider that the potential vector
field is pointwise collinear with the Reeb vector field and prove a
non-existence of such Cotton soliton. Next we assume that the potential vector
field is orthogonal to the Reeb vector field. It is proved that such a Cotton
soliton on a non-Kenmotsu almost Kenmotsu 3-$h$-manifold such that the Reeb
vector field is an eigen vector of the Ricci operator is steady and the
manifold is locally isometric to $\mathbb{H}^2(-4) \times \mathbb{R}$.
