Do we ever say the Hail Mary at Mass?

No, the Hail Mary is not a part of the Catholic Mass. That said, I have been to a few Masses where the priest inserts a Hail Mary at the end of his homily (this sometimes happens on Mother’s Day or on a Marian feast day). I know of no prohibitions against doing so as long as the Hail Mary does not replace one of the prayers of the Mass. Overall, though, the Hail Mary is prayed more as a private devotion or as part of the praying of the rosary.