How do I add a checkmark to my-sites save settings hook

It sounds as though have altered the page to submit your data. Assuming that that code work properly:

1. my-sites.php loads admin.php

   10  require_once( dirname( __FILE__ ) . '/admin.php' );
   

2. Which fires the load-{pagenow} hook

   330         /**
   331          * Fires before a particular screen is loaded.
   332          *
   333          * The load-* hook fires in a number of contexts. This hook is for core screens.
   334          *
   335          * The dynamic portion of the hook name, `$pagenow`, is a global variable
   336          * referring to the filename of the current page, such as 'admin.php',
   337          * 'post-new.php' etc. A complete hook for the latter would be
   338          * 'load-post-new.php'.
   339          *
   340          * @since 2.1.0
   341          */
   342         do_action( 'load-' . $pagenow );
   

That is probably the hook you need, but based on the very generic information you’ve provided it is hard to write much code.