Output single post custom taxonomy child terms

I hope I understand your question. You are looking for all terms connected to a single post, but only terms that are children to the adoption-status term

To get all the terms from a specific post, you can make use of wp_get_post_terms(). This will return all the terms of a specific taxonomy connected to the post.

You will also need to make use of get_term_children() to get all the child terms from your adoption-status term

Once you have all that, it is simply to compare the two, and only returning the terms that are associated with the post and is a child term to your given term.

You can try something like the following (NOTE: Change the term ids and taxonomy names accordingly)

$child_terms = get_term_children(21, 'category');
$all_terms   = wp_get_post_terms($post->ID, 'category');

foreach ( $all_terms as $term ) {

    if( !in_array($term->term_id, $child_terms ) ) 
        continue;

    echo $term->name;

}