Pagination with custom field

As I stated in comments, you need pre_get_posts and a meta_query to exclude posts from a certain custom field with a certain value from the main query. It is really in your best interest to read up on the before mentioned. Just remember, all parameters in WP_Query works on pre_get_posts as WP_Query uses pre_get_posts to alter the query parameters, and the main query uses WP_Query

In short, you can try the following: (REQUIRES PHP 5.4+ and the code is untested)

add_action( 'pre_get_posts', function ( $q )
{
    if (     !is_admin() // Targets the front end only, we don't need this for is_home()
          && $q->is_home() // Change this to the appropriate page, here we only target the home page
          && $q->is_main_query() // Target the main query only
    ) {
        $meta_query = [
            [
                'key' => 'Custom_ID',
                'compare' => 'NOT IN' // Do not get posts from this spoecific key/value pair
            ]
        ];
        $q->set( 'meta_query', $meta_query );
    }
});