Let $f: X \to Y$ be a finite, surjective morphism of smooth, quasi-projective varieties over a field $k$ of characteristic zero. Let $p \in X$. Is it true that I can find a subvariety $X' \subseteq X$ of codimension one such that $p$ is a smooth point of $X'$ and such that $f(p)$ is a smooth point of the closure of $f(X')$ in $Y$?
Edit: Alternatively: If $\dim X >0$, can I find a smooth curve $C$ on $X$ such that $f(p)$ is a smooth point of the closure of $f(C)$ in $Y$?