Please recommend me some manga where the MC is abused by their family and decides to move ...

First story on this one has what you're looking for http://www.mangago.me/read-manga/everyday_smile/

This one has something similar but it's not the main plot http://www.mangago.me/read-manga/tasogare_outfocus/

This here is super dark and has a sequel but it worth reading http://www.mangago.me/read-manga/sumomo_harada/

Thank you!