Add classes to add_submenu_item menu items in WordPress

add-submenu-item-classes.php

<?php

add_action( 'admin_menu', function() {

    $parent_slug = 'options-general.php'; // customize to suit
    $menu_slug   = 'my-submenu-item';     // customize to suit
    $position    = 6; 		      // customize to suit

    add_submenu_page(
        $parent_slug,
        __( 'Submenu Item', 'my-submenu-item' ),
        __( 'Submenu Item', 'my-submenu-item' ),
        'manage_options',
        $menu_slug,
        'render_submenu_page',
        $position
    );

    // after calling add_submenu_page() your sub menu page
    // is now available on the global $submenu array...
    global $submenu;

    // if you don't supply a $position arg to add_submenu_page() or you are targeting
    // submenu items you have no or limited control over, you can find the positional
    // index by searching the submenu context for the parent slug by your menu slug.
    // UNCOMMENT THE NEXT TWO LINES TO SEARCH DYNAMICALLY BY MENU SLUG
    // $position = array_search( $menu_slug, array_column( $submenu[ $parent_slug ], 2 ) ); // get the offset (indexes are numerical strings)
    // $position = key( array_slice( $submenu[ $parent_slug ], $position, 1, true ) );      // find the actual $position assigned by core

    // add your classes to the fourth index of the submenu item for the $parent_slug
    // at your $menu_slug $position, where this index holds element classes, this will
    // add the same classes to the outer <li> submenu item and child <a> element
    // @see _wp_menu_output() in wp-admin\menu-header.php:L199
    $submenu[ $parent_slug ][ $position ][4] = "{$menu_slug} dashicons-before dashicons-admin-settings";

});

add_action( 'admin_head', function() {

    ?>

    <style>

        /** hide the dashicons-before:before attributes on li element */
        ul.wp-submenu li.my-submenu-item.dashicons-before:before {
            display: none;
        }

        /** adjust to suit your preferences here if required */
        ul.wp-submenu a.my-submenu-item.dashicons-before:before {
            margin-right: 5px;
        }
    </style>

    <?php }

);

Alternatively you can target the anchor href attribute using pure css alone:

ul.wp-submenu a[href^="options-general.php?page=my-submenu-menu-slug"]:before {
    margin-right: 5px;
    font-family: dashicons;
    display: inline-block;
    line-height: 1;
    font-weight: 400;
    font-style: normal;
    speak: never;
    text-decoration: inherit;
    text-transform: none;
    text-rendering: auto;
    -webkit-font-smoothing: antialiased;
    -moz-osx-font-smoothing: grayscale;
    width: 20px;
    height: 20px;
    font-size: 20px;
    vertical-align: top;
    text-align: center;
    transition: color 0.1s ease-in;
    content: "\f180";
}