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

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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.