A note on coverings with special fibres and monodromy group $S_{d}$

We consider branched coverings of degree $d$ over $Y$ with monodromy group $S_{d}$, $k$ points of simple branching, $n-k$ special points and fixed branching data at the special points, where $Y$ is a smooth connected complex projective curve of genus $g\geqslant1$, and $n$, $k$ are integers with $n>k>0$. We prove that the corresponding Hurwitz spaces are irreducible if $k>3d-3$.