Add content under the search form (mobile fullscreen)

Hello,

Yes, it is possible. But only with help of some custom js code.
Please use code snippet

add_action( 'wp_head', 'my_wp_head' );

function my_wp_head() { ?>

    <script>

        window.addEventListener('load', function() {

            jQuery(document).on( 'focus', '.aws-mobile-fixed .aws-search-field', function() {
                var container = jQuery(this).closest('.aws-mobile-fixed');
                var customContent = 'Some content';
                container.after('<div class="aws-custom-content" style="position:fixed!important;z-index: 9999999 !important;top: 120px !important;left: 20px !important;">'+customContent+'</div>');
            });

            jQuery( document ).on( 'click', '.aws-mobile-fixed-close', function(e) {
                jQuery('.aws-custom-content').remove();
            });

        }, false);

    </script>

<?php }

You need to add it somewhere outside plugins folder. For example, inside functions.php file of your theme or use some plugin for adding code snippets.

Regards