Suppose that $X$ and $Y$ are discrete random variables and let $A$ be an event. I'm trying to figure out which one of the two definitions is the correct one for the conditional entropy, as this isn't clear to me from the standard definition (which doesn't consider conditioning on additional events). It seems to me that either of the following two could be the right interpretation: $$ H[ X \mid Y,A ] = \sum_y p(y) H[ X \mid Y=y,A], $$ $$ H[ X \mid Y,A ] = \sum_y p(y \mid A) H[ X \mid Y=y,A]. $$
I'm leaning towards the latter definition because the conditioning on event $A$ restricts the joint probability space from which we are sampling $X$ and $Y$. However, I can't formalize this intuition and, as mentioned, I'm not even sure if it is correct!