  A toric del Pezzo surface $X_P$ with cyclic quotient singularities determines
and is determined by a Fano polygon $P$. We construct an affine manifold with
singularities that partially smooths the boundary of $P$; this a tropical
version of a Q-Gorenstein partial smoothing of $X_P$. We implement a mild
generalization of the Gross-Siebert reconstruction algorithm - allowing
singularities that are not locally rigid - and thereby construct (a formal
version of) this partial smoothing directly from the affine manifold. This has
implications for mirror symmetry: roughly speaking, it implements half of the
expected mirror correspondence between del Pezzo surfaces with cyclic quotient
singularities and Laurent polynomials in two variables.
