  We study topological recursion on the irregular spectral curve $xy^2-xy+1=0$,
which produces a weighted count of dessins d'enfant. This analysis is then
applied to topological recursion on the spectral curve $xy^2=1$, which takes
the place of the Airy curve $x=y^2$ to describe asymptotic behaviour of
enumerative problems associated to irregular spectral curves. In particular, we
calculate all one-point invariants of the spectral curve $xy^2=1$ via a new
three-term recursion for the number of dessins d'enfant with one face.
