Is it proper to pray the Hail Mary during Mass?

I converted to the Catholic Church about 4 years ago. Our parish recently had a priest from Africa join us as our second priest. After the Prayers of the Faithful, he consistently has everyone pray a Hail Mary for Mary’s intercession. This is the first time I have seen this and I am unsure about this practice. Don’t get me wrong. I have great admiration and love for our Blessed Mother and frequently pray the Rosary with a group of fellow parish members before daily Mass. The priest seems to be a very devout man and I have great respect for him and he has heard my confessions on several occasions. I have not heard anyone else expressing their concern about this. I certainly do not want to rock the boat by asking questions of other parish members, but I do wish to keep the Liturgy in tact and in conformity with Church instructions. Is this practice consistent with Catholic Liturgy guidelines?

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Should the Hail Mary be prayed at Mass?
Why can’t the Hail Mary be prayed at Mass?

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