I have visited your site and core plugin is updated. You can update Royal core via WP Dashboard > Updates > click Check again button and then choose plugin to update. Or manually via FTP using plugin archive from the Downloads page https://www.8theme.com/downloads
But 2 required plugins were not installed http://prntscr.com/rs4d37
Always update theme and plugin in a bundle. Don’t update just theme or just plugin.
Also, we would recommend you to use the staging site to test update. Then if everything is ok update the production site. In this case your production site won’t be down and you won’t lose any client.
Is there anything still wrong at your site?