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Given a tagged triangulation of a once-punctured polygon $P^*$ with $n$vertices, we associate an ice quiver with potential such that the frozen partof the associated frozen Jacobian algebra has the structure of a Gorenstein$K[X]$-order $\Lambda$. Then we show that the stable category of the categoryof Cohen-Macaulay $\Lambda$-modules is equivalent to the cluster category$\mathcal{C}$ of type $D_n$. It gives a natural interpretation of the usualindexation of cluster tilting objects of $\mathcal{C}$ by tagged triangulationsof $P^*$. Moreover, it extends naturally the triangulated categorification by$\mathcal{C}$ of the cluster algebra of type $D_n$ to an exact categorificationby adding coefficients corresponding to the sides of $P$. Finally, we lift theprevious equivalence of categories to an equivalence between the stablecategory of graded Cohen-Macaulay $\Lambda$-modules and the bounded derivedcategory of modules over a path algebra of type $D_n$.

Ice quivers with potential arising from once-punctured polygons and Cohen-Macaulay modules