The Harer--Zagier recursion for an irregular spectral curve

We derive the Do and Norbury recursion formula for the one-loop mean of an irregular spectral curve from a variant of replica method by Brez\'in and Hikami. We express this recursion in special times in which all terms $W_1^{(g)}$ of the genus expansion of the one-loop mean are polynomials. We find a generalization of this recursion to the generalized Laguerre polynomial case.