That would rub me the wrong way if I hadn't asked for it. I am just the sort of person who does not want pity or charity. If I want help, I'll ask for it or do the hiring, thanks.
In particular, mowing the lawn is one of my "escapes" where I can kind of zone out doing something that is routine and mindless for a little while. It's almost like meditation. Shoveling snow off of the driveway in the winter is another one.
So I would say, ask the person first. "I'd like to take care of the lawn for you if that would be helpful" rather than just doing it.