Yes, the governess should instead clean the poo poo.
I also think that's a kind of lesson for Anastasia to teach her dog (but it may be Ortino, too, i don't know)how to do it on the right place.
Alexandra wants her to clean the poo poo so that she would not let Shvibzk (or ortino) do it again. She knows that one would be the fairest thing to do.