This question is related to a question I already asked on MO (smooth quotient out of a singular variety?), but I realized later that the hypotheses where not precise enough in my former question.

Let $p : X \rightarrow Y$ be a finite morphism of algebraic varieties over $\mathbb{C}$ with $Y$ smooth and $X$ normal. By Abhyankar's theorem, we know that if the ramification locus of $p$ is a divisor with normal crossings, then $X$ has quotient singularities (and in particular has rational singularities).

I would like to know if there are weaker hypotheses on the ramification locus which would imply that $X$ is Cohen-Macaulay? One important thing is that I don't want to assume that $p$ is flat (in fact I want to prove that $p$ is flat).

Thanks!