Hi Patrick,
Well, that needs to be custom coded, so that the menu click triggers a focus on the input. Some search menu icons have that pre-programmed, for others a small code is needed. This one should cover your case:
add_action('wp_footer', 'asp_add_menu_focus_trigger');
function asp_add_menu_focus_trigger() {
?>
<script>
jQuery(function($){
$('.nav-search-link').on('click', function(){
$('.nav-search-form').find('input.orig').get(0).focus();
});
});
</script>
<?php
}
Try adding this custom code to the functions.php in your theme/child theme directory. Before editing, please make sure to have a full site back-up just in case!