I think the most important role the tutor can play is to really listen to the student and try to understand where they’re coming from with their writing. If you assume a student is going one direction when really they’re trying to go in a different direction neither of you will get anything out of working together. But by listening and allowing the student to talk, you’re also giving them the opportunity to be heard which is usually what writing is all about.
Another thing the tutor can do is ask questions that will help the student come to their own conclusions, while also sliding in subtle hints to improve the work without hijacking it. You have to remember this is their writing, not yours. It's easy to cross that line when you get caught up in the writing because you know the answer, but they will never learn anything if you just do it for them. They may need help coming up with the answers, but they should be the ones finding them.