Category permalinks don’t work even after flushing

The default WP category is only set-up to show posts out of the box. You should be able to expand this by doing something like the below in your functions.php:

// Order course archive and taxonomy views
function adjust_main_query($query){
    if ( ! is_admin() && is_category() && is_category() && $query->is_main_query() ) {

        $query->set( 'post_type', array( 'post', 'mpCPT' ));
    }
}
add_action( 'pre_get_posts', 'adjust_main_query' );

I haven’t tested this, but it’s an adaptation of a working example I have used. Hope it helps get you on the right lines.