The humeral veil.
The humeral veil is not only worn by the subdeacon. It is also occasionally worn by the priest; for example, when giving the Benediction of the Blessed Sacrament. The bishop’s assistants also wear a veil (a vimpa) when holding his mitre and crozier.
The veil is used a sign of respect when holding the sacred vessels.
In the particular case of the holding the paten, it is derived form the use of the acolyte holding the paten (touching of the sacred vessels being restricted to the higher orders) in the ancient liturgy of Rome, using a veil. The acolyte would bring the paten to the altar at the beginning of the Canon. He would then hand to an assisting subdeacon around the middle, who would hand it to one of the main subdeacons for the Mass, a “regional” subdeacon. At the end of the Canon this subdeacon would give it to the archdeacon you would give it to the deacon (sounds a bit like passing the parcel ). After the Pope had completed the fraction, the Body of Christ was placed on this paten. It would later be further fractioned and taken to the throne for the communion of the Pope.
I think the veil worn by the subdeacon during the Mass is connected to the holding of the paten- when the paten is not held (like in Masses of the Dead) the veil is not worn