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Thanks!

The issue on that case is, that the fixed header is covering up the search bar visibility. The problem with that, that it is not possible to detect, the plugin can only scroll to the window viewport. I’m afraid I cannot do anything about it.

You can try a custom function to croll after the animation completes, but I am not sure if it will do any good:

add_action('wp_footer', 'asp_custom_wp_footer');
function asp_custom_wp_footer() {
  ?>
  <script>
  jQuery(function($) {
    var t;
    $('.asp_main_container').on('asp_search_end', function(){
        clearTimeout(t);
        var $this = this;
        setTimeout(function(){
          $('body,html').scrollTop($('body,html').scrollTop() - 120)
        }, 740);
    });
  });
  </script>
  <?php
}

That i not the auto populate option, you can find it here.

Make sure to set the search settings position to block. Also, check if this option is disabled on the mobile settings.

If you want to forcefully center the block layout, you can use this custom CSS:

.asp_sb form {
    display: flex !important;
    justify-content: center;
}

Unfortunately that is possibly the only way. I am not seeing that custom code on the linked page, so there still might be hope. Make sure to place it into the functions.php file in your active theme/child theme directory. After that, clear all layers of cache so it populates to the front-end.