There's no right answer here in particular; mostly it depends on how your team agrees to use the GitHub PRs.
If you "Start a Review", then in addition to saving up all the comments & notifying the user once, as Nezih mentioned, GitHub will also require you to choose "Approve", "Request Changes", or "Comment". If you were a requested reviewer, it will then indicate that you have submitted a review (no matter what you pick). Leaving a single comment doesn't do that.
Because it saves up everything to notify the person only once when you are finished, it has occasionally happened to me that by the time I have finished my review, the code has already changed again as the author has responded to other change requests. That's confusing and annoying, and sometimes influences me to leave single comments instead. On the other hand, if I am going to make a lot of comments, I feel it is kinder to the author to batch them up in the "Start a review" mode. This is especially true if my comments are going to refer to each other.
If you are posting a refactored bit of code as a suggestion, you could use either option. If you don't think the PR should be merged without your suggestion, then choose "Start a Review" and select "Request Changes."
If your team has agreed that teammates can modify each other's code for things like that (which is unusual, but there are situations where it works well), then you would be checking out the branch and pushing your own commit to it. But make sure you understand the team norms first: in my team, we don't do that without asking the author first so as to avoid potential clobbering of work in progress.