Show single Child Category on Custom Post

You can use get_term_children function.

For Example:

<?php
$termID = 10;
$taxonomyName = "region";
$termchildren = get_term_children( $termID, $taxonomyName );

echo '<ul>';
foreach ($termchildren as $child) {
    $term = get_term_by( 'id', $child, $taxonomyName );
    echo '<li><a href="' . get_term_link( $term->name, $taxonomyName ) . '">' . $term->name . '</a></li>';
}
echo '</ul>';
?>