Yes it does make a difference.
The further they got into it, the more likely they would have to play some of their first team players.
It also gives them something else to focus on and not just the PL. I would not want them to win it just get to the quarter finals or maybe the semi finals and then lose.
I hope they do not go out of the FA cup in the 3rd round either.
I want them in the Thursday night crap next season or worse but I'm sure they will be in Europe next season.
I DON'T want them in the CL next season.