This snippet (which goes into your functions.php) allows you to apply styles after someone clicked on the mobile menu (e.g. change the background color of the header).

functions-mobile_nav-opened.php

// This snippet adds an event Listener to the mobile nav menu and if someone opens it, the body tag will receive the additional class "mobile_nav-opened".
function nxt_header_js_code() {
	?>
	<script>
		const mob_nav = document.getElementsByClassName('mobile_nav')[0];
		// console.log(mob_nav);
		mob_nav.addEventListener('click', function() {
			document.getElementsByTagName('body')[0].classList.toggle('mobile_nav-opened');
		});
	</script>
	<?php
}
add_action('wp_footer', 'nxt_header_js_code');