Let $X,Y,Z$ be projective $3$-folds. Assume that $Y$ is smooth and $Z$ is smooth and Fano. Moreover, assume that there is a generically finite morphism $f:Y\rightarrow Z$ admitting a factorization $f=h\circ g$ where $g:Y\rightarrow X$, $h:X\rightarrow Z$ are two generically finite morphism.
Can we say something about the singularities of $X$ or could it have arbitrarily bad singularities ?
For instance, does the assumptions imply that $X$ has at worst canonical singularities ?