  For a closed oriented surface $ \Sigma $ let $X_{\Sigma,n}$ be the space of
isomorphism classes of orientation preserving $n$-fold branched coverings $
\Sigma\rightarrow S^2 $ of the two-dimensional sphere. At a previous paper, the
authors constructed a compactification $\bar{X}_{\Sigma,n}$ of the space that
coincides with the Diaz-Edidin-Natanzon-Turaev compactification of the Hurwitz
space $H(\Sigma,n)\subset X_{\Sigma,n}$ consisting of isomorphism classes of
branched coverings with all critical values being simple. Using Grothendieck's
dessins d'enfants we construct a cell structure of the compactification. The
obtained results are applied to the space of trigonal curves on a Hirzebruch
surface.
