Hi,
I believe the only option there is to use a custom code, since the plugin is integrated into a custom container.
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!
add_action('wp_footer', 'asp_focus_on_menu_click');
function asp_focus_on_menu_click() {
?>
<script>
jQuery(function($){
$('li.fusion-main-menu-search a').on('click', function(){
$('.fusion-custom-menu-item-contents .asp_m input.orig').get(0).focus();
});
});
</script>
<?php
}
This should force the input focus, once the magnifier icon is clicked.