I am asking mostly for reference if such a definition exists in the literature. I am also interested in the count if it appears somewhere.

Let $\mu:=(\mu_1,\ldots , \mu_n)\vdash d$ for positive integer $d$ then simple Hurwitz number $H_{g}(\mu)$ is defined to be the number of degree $d$ covering map say $f$ counted with weight $\frac{1}{m!\, Aut(f)}$ from $\bf{smooth} $ genus $g$ surface to $\mathbb{CP}^1$ having $m$ simple branch point and branch point of type $\mu$ at $\infty$.

$m2=g-2+n+d$ denote the number of simple singularities by Riemann Hurwitz formula.

Let me give a few counting of these numbers. $$H_0 (3)= \frac32$$ $$ H_1 (3)=\frac98$$ $$ H_0 (4)=\frac83$$

If there is in literature a definition of Hurwitz number that will extend the above definition of Hurwitz number from a nodal curve of genus $g$ to $CP^1$. Also, the weighted count would be finite. We need to take into account the nodal points.