I think Aya is studying so she can go to a school in America so she can be with Mitsuki… I think I may be wrong but I remember that being a thing….?
I think she just wants to understand lyrics of songs she listens to. People keep bringing up Mitsuki going to USA, but I personally feel that particular plot point was shot down almost on arrival.
In manga itself, the only person who brought up that idea was Kanna, in a half-joking way, to tease Joe. She did not bring this up again ever again, so even if we consider she was serious about it, she seems to have given up on the idea.
Mitsuki herself was never exposed to the idea of going to USA, she haven't ever bring up any desire to go, nor anybody else toted that idea to her. And, honestly, as much as I can read her personality, I don't think she is a type of person to leave everything she knows in pursuit of some ambitious musical career.
The only reamining plot point that I can see that would lead Mistuki to leave country is to follow her uncle, and I honestly think it would go against their character arc we were shown. Joe wants to pursuit his own happiness, Mitsuki wants it to without keeping him from pursuit it, if she was to follow him, I feel that it would be against these themes.
And why does she need to leave if she wanted to pursue musical career? We already have someone from industry introduced in the person of Hime's father, I find it more likely this is where Mitsuki's way will lead her if she is to go into show biz.
Hell, why does she need to go the musical industry road? The comic is taking place in the modern times, with social media and the internet in full swing, so she can pretty much become a famous singer from anywhere in the world. Hell, she already went viral, why not pursue this path of indie artist?
Not even mentioning that American Dream is dead and there is no more chance to get successful for anyone in USA than anywhere else in the world, Or that modern times USA is not really that safe to travel for foreigners.