Ignore/Skip default value on orderby menu_order?

You can try the following (untested) mini plugin:

<?php
/**
 * Plugin Name: Support for ignoring the default menu order in WP_Query
 * Description: Uses the _ignore_default_menu_order argument
 * Plugin URI:  http://wordpress.stackexchange.com/a/193291/26350
 */
add_filter( 'posts_where', function( $where, $q )
{
    global $wpdb;

    if( (bool) $q->get( '_ignore_default_menu_order' ) ) {
        $where .= "AND {$wpdb->posts}.menu_order <> 0";
    }
    return $where;

}, 10, 2 );

Then you should be able to use the new custom query argument like:

$query = new WP_Query( 
    [ 
        '_ignore_default_menu_order' => true,
    ]
);

to ignore posts with the default menu order (0).

You could also extend this to support any menu order as user input.