Using views, how to render a block of father terms with a brief shortlist of child terms?
If you didn't find better solution you may create one view that will display parent term name and parent tid and second view that will display four children with a contextual filter by parent tid. And using template_preprocess_views_view_field insert view with terms children in the view with parents.

/**
 * Implements hook_preprocess_views_view_field().
 */
 function mymodule_preprocess_views_view_field(&amp;$vars) {
   switch ($vars['view']-&gt;name) {
     case 'terms_parents':
       if ($vars['field']-&gt;options['id'] == 'tid') {
         $vars['output'] = views_embed_view('children_view', 'display', array($vars['output']));
       }
       break;
   }
}