  Given a surface cyclic quotient singularity $Q\in Y$, it is an open problem
to determine all smoothings of $Y$ that admit an anticanonical model and to
compute it. In [HTU], Hacking, Tevelev, and Urz\'ua studied certain irreducible
components of the versal deformation space of $Y$, and within these components,
they found one parameter smoothings $\mathcal{Y} \to \mathbb{A}^1$ that admit
an anticanonical model and proved that they have canonical singularities.
Moreover, they compute explicitly the anticanonical models that have terminal
singularities using Mori's division algorithm [M02]. We study one parameter
smoothings in these components that admit an anticanonical model with canonical
but non-terminal singularities with the goal of classifying them completely. We
identify certain class of "diagonal" smoothings where the total space is a
toric threefold and we construct the anticanonical model explicitly using the
toric MMP.
