May must be either Aurora or finch. This is a one sided exclusive or involving the place and bird categories, so we look at the right side to see what places are possible and what birds are possible.
Let's check the places first.
The right side can either be Aurora (given in the clue) or any of the places associated with the finch.
We already know that for clues like this, the two options on the right side of the clue cannot be in the same row. So Aurora cannot be finch.
From your grid or the other clues, the finch must have been excluded from Yetter to get the hint you indicated.
So if the finch cannot be from Yetter, how could May ever be associated with Yetter?
Aurora isn't Yetter, and if the finch cannot be from Yetter, then there is no way for May to be associated with Yetter.
To answer your question:
"Finch maybe totally unrelated to MAY so why should anything to do with Finch affect MAY at all?"
The clue says that the finch and May could be in the same row. That's one of the two possibilities.
If May is in the same row as Aurora, then we don't care where the finch is, and, as you indicate, the finch doesn't affect May.
However, if May isn't in the same row as Aurora, then according to the clue, May must be in the same row as the finch. And if the finch cannot be in the same row as Yetter, then May and the finch have to be in a row that is different from the one that Yetter is in.
I think this explains your main question, so let's continue the process to squeeze as many Xs out of that clue as we can.
It might not be necessary, but while you're checking the right side of that clue for the place exclusions (like Yetter), check it for the bird exclusions too.
There might be some birds excluded from Aurora in addition to the finch.
For example, if the parrot cannot be from Aurora, then the right side of the clue cannot include the parrot. So, for this made up example, you could deduce that May cannot be the parrot.
Visually, this whole process can be accomplished by looking at the intersection of Aurora and finch. You will be placing an X at that intersection, and while you're there, check the row and column of the intersection for extra Xs that you can transpose to the May row.