It's actually rather simple.
1. Go into your wordpress admin interface and edit the WordPress Address (URL) AND site URL to reflect what you want it to become (cyfairnwoa.x10.mx). Backup your database first to ensure you can restore functionality if something goes wrong.
2. Move (or copy to be safe until confirmed it works) all files from public_html/wp to public_html replacing any files that are already there (This assumes you do not want anything about the old file to remain online). You have now moved your wordpress to load up from the url of your old site.