You should discuss an alternative scheme for the homework problems for this student. Just tell the student that the way things are going now is not acceptable, he'll not get the credits for the course. If he claimstells you that he'll try harder, tell him that you actually know why he isn't doing well on the assignments, that he is going to fail the course and may also become ill due to overwork and then risk dropping out of university altogether.
Then offer him an alternative scheme for the assignments that is more compatible with his schedule, but which also comes with zero tolerance for not adhering to it. You should aim for assignments for him that are at least as hard as the regular assignments. You should tell in class that some students are following an alternative scheme for assignments to accommodate for private issues, that this is open to everyone, but the assignments are on average a bit harder than the regular assignments. This makes sure all students are treated equally, and that the alternative scheme is not an easy pass so the floodgates for the alternative scheme are not opened.