  We consider maps on genus-$g$ surfaces with $n$ (labeled) faces of prescribed
even degrees. It is known since work of Norbury that, if one disallows vertices
of degree one, the enumeration of such maps is related to the counting of
lattice point in the moduli space of genus-$g$ curves with $n$ labeled points
and is given by a symmetric polynomial $N_{g,n}(\ell_1,\ldots,\ell_n)$ in the
face degrees $2\ell_1, \ldots, 2\ell_n$. We generalize this by restricting to
genus-$g$ maps that are essentially $2b$-irreducible for $b\geq 0$, which
loosely speaking means that they are not allowed to possess contractible cycles
of length less than $2b$ and each such cycle of length $2b$ is required to
bound a face of degree $2b$. The enumeration of such maps is shown to be again
given by a symmetric polynomial $\hat{N}_{g,n}^{(b)}(\ell_1,\ldots,\ell_n)$ in
the face degrees with a polynomial dependence on $b$. These polynomials satisfy
(generalized) string and dilaton equations, which for $g\leq 1$ uniquely
determine them. The proofs rely heavily on a substitution approach by Bouttier
and Guitter and the enumeration of planar maps on genus-$g$ surfaces.
