Show child taxonomies (that have posts) of the current parent taxonomy

Props to Sally for improving on this to show post count

$term = get_queried_object();

$children = get_terms( $term->taxonomy, array(
    'parent'    => $term->term_id,
    'hide_empty' => false
) );

if ( $children ) { 
    foreach( $children as $subcat )
    {
        echo '<li><a href="' . esc_url(get_term_link($subcat, $subcat->taxonomy)) . '">' . // wrapped
          $subcat->name . ' (' . $subcat->count . ')' . '</a></li>';
    }
}

UPDATE

To show the posts count:

As pointed in the comment, and based on the above code, you can use $subcat->count to display the number of posts in the specific term.

Therefore I replaced the $subcat->name . '</a></li>' with:

$subcat->name . ' (' . $subcat->count . ')' . '</a></li>'

which outputs something like: Term Name (1), Another Term Name (0), etc.

For the original code in question (using wp_list_categories()), here’s how to fix it, and the fixed code:

1. Replace the is_category() with is_tax( 'p_scales' ).

2. Replace the get_query_var( 'cat' ) with get_queried_object_id().

   if (is_tax('p_scales'))
   {
       $cur_cat = get_queried_object_id();
       if ($cur_cat)
       {
           $new_cats = wp_list_categories('show_option_none=&echo=false&child_of=" . // wrapped
             $cur_cat . "&depth=1&title_li=&show_count=1&hide_empty=1&taxonomy=p_scales');
           echo '' . $new_cats . '';
       }
   }