Let $f:X\to Y$ be a finite morphism between smooth projective varieties of the same dimension over $\mathbb C$. Then we have the following equality on canonical divisors $$K_X \cong f^* K_Y + R$$ where $R$ is an effective divisor supported on the ramification locus. In particular, we have
if $h^0(K_Y)\neq 0$, then $h^0(K_X)\neq 0$.
And my question is
if we generalize the condition "finite morphism" to "generically finite morphism", does this implication still hold?
At first I expect some formula like $$K_X \cong f^* K_Y + R +E$$ to be true, where $E$ is some effective divisor, and this will lead to what I want. But then I realized that this is not likely to be true when the exceptional fibers (i.e. those with positive dimension) do not form a divisor. Then I do not know how to approach this in general. Maybe this is just false but could anyone provide a counter-example?
Thanks in advance.