Change your wp_localize_script-Call to the right Action:
wp_localize_script( 'addItemToNav', 'menuItems', array(
// URL to wp-admin/admin-ajax.php to process the request
'ajaxurl' => admin_url( 'admin-ajax.php' ),
// generate a nonce with a unique ID "myajax-post-comment-nonce"
'postCommentNonce' => wp_create_nonce( 'update_nav_items-nonce' ),
'action' => 'wp_ajax_update_nav_items'
)
);
As ‘action’ you currently Input the function Name, but that is wrong. Your registered Action to call the function has an different Name.