Judging by the code for that like system, you should be able to find out if the user has liked the post like this:
if ( is_user_logged_in() ) {
$likes = get_user_meta( get_current_user_id(), '_liked_posts', true );
if ( is_array( $likes ) && in_array( get_the_ID(), $likes ) ) {
echo '<span class="liked">Liked</span>';
}
}
That will echo a <span>
with the class liked
if the user has liked the post. That code will need to go in the template where you want the “Liked” badge to appear.