Get 1st child of category only

At the moment, I'm using get_term_children() to get the children of a category. The problem I'm faced with now is that I only want the first child. I assume I would use get_terms() instead, but I've been battling to implement it with my current code. If someone could point me in the right direction, I'd greatly appreciate it:

    // get children of current parent.
    $tchildren = get_term_children($term->term_id, $taxonomy);

    $children = array();
    foreach ($tchildren as $child) {
        $cterm = get_term_by( 'id', $child, $taxonomy );
        // If include array set, exclude unless in array.
        if ( is_array( $include ) && ! in_array( $cterm->slug, $include ) ) continue;
        $children[$cterm->name] = $cterm;
    }

and the whole section of code is:

function my_dropdown_categories( $taxonomy, $current_selected = '', $include = null ) {
// Get all terms of the chosen taxonomy
$terms = get_terms($taxonomy, array('orderby' => 'name'));

// our content variable
$list_of_terms = '<select id="location" class="selectboxSingle" name="location">';

if ( ! is_wp_error( $terms ) ) foreach($terms as $term){

// If include array set, exclude unless in array.
if ( is_array( $include ) && ! in_array( $term->slug, $include ) ) continue;

$select = ($current_selected == $term->slug) ? "selected" : ""; // Note: ==

if ($term->parent == 0 ) {

    // get children of current parent.
    $tchildren = get_term_children($term->term_id, $taxonomy);

    $children = array();
    foreach ($tchildren as $child) {
        $cterm = get_term_by( 'id', $child, $taxonomy );
        // If include array set, exclude unless in array.
        if ( is_array( $include ) && ! in_array( $cterm->slug, $include ) ) continue;
        $children[$cterm->name] = $cterm;
    }
    ksort($children);

    // OPTGROUP FOR PARENTS
    if (count($children) > 0 ) {
    //     $list_of_terms .= '<optgroup label="'. $term->name .'">';
         if ($term->count > 0)
             $list_of_terms .= '<option class ="group-result" value="'.$term->slug.'" '.$select.'>' . $term->name .' </option>';
    } else
        $list_of_terms .= '<option value="'.$term->slug.'" '.$select.'>'. $term->name .' </option>';
    //$i++;

    // now the CHILDREN.
    foreach($children as $child) {
         $select = ($current_selected == $child->slug) ? "selected" : ""; // Note: child, not cterm
         $list_of_terms .= '<option value="'.$child->slug.'" '.$select.'>'. $child->name.' </option>';

    } //end foreach

    if (count($children) > 0 ) {
        $list_of_terms .= "</optgroup>";
    }
}
}

$list_of_terms .= '</select>';

return $list_of_terms;
}

add_action( 'wp_ajax_wpse158929_get_terms_for_cpt', 'wpse158929_get_terms_for_cpt' );
add_action( 'wp_ajax_nopriv_wpse158929_get_terms_for_cpt', 'wpse158929_get_terms_for_cpt' );