What is the purpose of the cover the priest has over the Chalice during the Eucharistic Prayer?
The “cover” is called the Pall. It has been used for centuries. The purpose for it was to prevent dust or other foreign objects from falling in the Chalice. Until after the reforms following the Second Vatican Council, it was a liturgical requirement that the Pall be kept on the chalice except during the Consecration and Communion. That requirement was abolished. So it is now a practical decision for the priest to make. Some priests like to adopt the older practice.
