In this sentence, it's saying "Father is busy writing (or writing letters)." But it's using 正 and 着 to indicate that it's currently happening (in progress right now). Does "父亲正忙着写信" have ANY difference whatsoever from "父亲在忙写信"?
Thanks so much! I'm sorry for asking so many questions, but this is such an amazing resource, and I feel like asking these questions only helps the community and fellow students of this beautiful language.