356. In choosing texts for the different parts of the Mass, whether for the time of the year or for Saints, the norms that follow should be observed.
357. Sundays and Solemnities have assigned to them three readings, that is, from a Prophet, an Apostle, and a Gospel, by which the Christian people are instructed in the continuity of the work of salvation according to God’s wonderful design. These readings should be followed strictly. In Easter Time, according to the tradition of the Church, instead of being from the Old Testament, the reading is taken from the Acts of the Apostles.
For Feasts, two readings are assigned. If, however, according to the norms a Feast is raised to the rank of a Solemnity, a third reading is added, and this is taken from the Common.
One example of a feast being elevated to a solemnity would be the parish patronal feast. And the reading, obviously, would be added to complete the Old Testament/New Testament part of the trio to accompany the Gospel.
For Memorials of Saints, unless proper readings are given, the readings assigned for the weekday are normally used. In certain cases, particularized readings are provided, that is to say, readings which highlight some particular aspect of the spiritual life or activity of the Saint. The use of such readings is not to be insisted upon, unless a pastoral reason truly suggests it.
So there is some flexibility on a reading associated with a saint. When might such a reading “be insisted upon”? Maybe when there is an image of the saint in the parish–a window, perhaps. And there is an accompanying seed of devotion connected to that image.