A morphism of curves is said to be Galois if the corresponding extension of function fields is Galois.

Let $f:X\longrightarrow Y$ be a finite morphism of smooth projective connected curves over $\mathbf{C}$ (or Riemann surfaces if you prefer) with branch locus $B\subset Y$ and ramification locus $R\subset X$. There exists a finite morphism $g:W\longrightarrow X$ of smooth projective curves such that the composition $f\circ g:W\longrightarrow Y$ is Galois. Take the Galois closure for example.

Can we say something about the branch locus of $g$? For example, when does it lie in $R$?

In my problem I would like to be able to choose $g$ such that the branch locus of $g$ lies in $R$. I have a feeling this is not always possible unfortunately.

The following might help actually. In my problem, $Y$ is the projective line and $f$ is a Belyi morphism. Thus, what I would like to see is that I can choose $g:W\longrightarrow X$ such that the composition $f\circ g$ is Belyi and Galois. Again, this seems unlikely to be possible.