Don't worry about this. You are thinking too much
In general, if you are talking about a particular mouse, the mouse, in specific cup, so use "the"
Better to say, "There's a mouse in that cup." ((specific mouse in specific cup)
However, if you are speaking generally about the idea of a mouse in any one of the drawers, you could use, "a" Imagine this conversation.
What wrong, Mary?
There's a mouse in a drawer and it scared me. Then I saw another in a cup.
Hope you aren't confused, because I am, now :)
PS: There're is not seen in writing . Use there are