I am just cleaning up the threads showing in "Show new replies to your posts."
You can empty the list quickly by "mark all read" at the right bottom. That way, all threads vanish up until a new reply is posted.
How can I determine which threads are showing in my "Show new replies to your posts."?
All threads in which you have posted, and which have replies you haven't read or marked as read yet will show up in this list.
To remove a thread permanently, you'd have to delete all your posts in that thread.
I originally thought that ones can determine it with watch and unwatch.
Those are for adjusting your
watchlist (linked right below the replies), not the list of new replies.