I’ve set up an extra field for all the available widgets. How do I show it’s value in the front-end?

Won’t a filter on widget_title do this?

function widget_title_wpse_206841($title,$instance) {
  // var_dump($instance);
  if (!empty($instance['subtitle'])) {
    $title .= ' something '.$instance['subtitle'].' something ';
  }
  return $title;
}
add_filter('widget_title','widget_title_wpse_206841',10,2);