Show recent posts from a custom taxonomy in wordpress

What you need is to add a tax_query to your WP_Query arguments, where you check that your taxonomy exists on the posts you are querying.

Here is an example that fetches the 3 latest published posts that have the taxonomy site set.

$args      = array(
    'post_type'      => 'post',
    'post_status'    => 'publish',
    'posts_per_page' => 3,
    'tax_query'      => array(
        array(
            'taxonomy' => 'site',
            'operator' => 'EXISTS',
        ),
    ),
);
$the_query = new WP_Query( $args );

Good reads

• Documentation for WP_Meta_Query
• Documentation for WP_Query