Conditional label Woocommerce archive [unsolved]

You need to access the Woo product loop, to output extra content on shop/archive pages (unless you use a custom template)

Try something like this:

add_action( 'woocommerce_after_shop_loop_item', 'conditional_label', 5 );
 
function conditional_label() {
   global $product;
   if (has_term( 'new', 'product_cat', $product->get_id())) {
        echo '<p class="new-lable">New</p>';
   }
}

Not tested, but should get you in the right direction.

Have a look at this hook guide, for a visualization on where you can add the text.