  We prove quasi-polynomiality for monotone and strictly monotone orbifold
Hurwitz numbers. The second enumerative problem is also known as enumeration of
a special kind of Grothendieck's dessins d'enfants or $r$-hypermaps. These
statements answer positively two conjectures proposed by Do-Karev and
Do-Manescu. We also apply the same method to the usual orbifold Hurwitz numbers
and obtain a new proof of the quasi-polynomiality in this case.
  In the second part of the paper we show that the property of
quasi-polynomiality is equivalent in all these three cases to the property that
the $n$-point generating function has a natural representation on the $n$-th
cartesian powers of a certain algebraic curve. These representations are the
necessary conditions for the Chekhov-Eynard-Orantin topological recursion.
