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Simple Hurwitz numbers enumerate branched morphisms between Riemann surfaceswith fixed ramification data. In recent years, several variants of this notionfor genus $0$ base curves have appeared in the literature. Among them areso-called monotone Hurwitz numbers, which are related to the HCIZ integral inrandom matrix theory and strictly monotone Hurwitz numbers which count certainGrothendieck dessins d'enfants. We generalise the notion of Hurwitz numbers tointerpolations between simple, monotone and strictly monotone Hurwitz numbersto any genus and any number of arbitrary but fixed ramification profiles. Thisyields generalisations of several results known for Hurwitz numbers. When thetarget surface is of genus one, we show that the generating series of theseinterpolated Hurwitz numbers are quasimodular forms. In the case that allramification is simple, we refine this result by writing this series as a sumof quasimodular forms corresonding to tropical covers weighted by Gromov-Witteninvariants. Moreover, we derive a quantum curve for monotone and Grothendieckdessins d'enfants Hurwitz numbers for arbitrary genera and one arbitrary butfixed ramification profile. Thus, we obtain spectral curves via thesemiclassical limit as input data for the CEO topological recursion.Astonishingly, we find that the CEO topological recursion for the genus $1$spectral curve of the strictly monotone Hurwitz numbers compute the monotoneHurwitz numbers in genus $0$. Thus, we give a new proof that monotone Hurwitznumbers satisfy CEO topological recursion. This points to an unknown relationbetween those enumerants. Finally, specializing to target surface$\mathbb{P}^1$, we find recursions for monotone and Grothendieck dessinsd'enfants double Hurwitz numbers, which enables the computation of therespective Hurwitz numbers for any genera with one arbitrary but fixedramification profile.

Triply mixed coverings of arbitrary base curves: Quasimodularity, quantum curves and a mysterious topological recursions