How do you change the comment count in the back end posts list, to reflect unapproved comments, rather than all comments?

Add following code in your theme’s functions.php.

// To add extra column in column headers
add_filter('manage_posts_columns', 'bs_event_table_head');
function bs_event_table_head( $defaults ) {
    $defaults['pending_comments']  = 'Pending Comments';
    return $defaults;
}

// To add data in column for each post.
add_action( 'manage_posts_custom_column', 'bs_event_table_content', 10, 2 );
function bs_event_table_content( $column_name, $post_id ) {

    if ($column_name == 'pending_comments') {

        // getting comments' counts
        $comments_count = wp_count_comments();
        $url = admin_url('edit-comments.php?comment_status=moderated&p=');
        echo '<a href="'.$url.$post_id.'" title="'.$comments_count->moderated.' pending" class="post-com-count"><span class="comment-count">'.$comments_count->moderated.'</span></a>';

    }

}

// To make column sortable
add_filter( 'manage_edit-post_sortable_columns', 'bs_event_table_sorting' );
function bs_event_table_sorting( $columns ) {
    $columns['pending_comments'] = 'pending_comments';
    return $columns;
}