From this, Rochester diocese ends illicit practice of allowing lay people to preach homilies, perhaps because this was going on for such a long time, a person would ask what is the authoritative Catholic Church Teaching that prohibits a lay person (of either gender) from preaching the homily at Mass and are there cases where the laity may (men or women) can be allowed to preach in a church or oratory? Under what circumstances?
Code of Canon Law states:
766.0 Lay persons can be permitted to preach in a church or oratory, if necessity requires it in certain circumstances or it seems advantageous in particular cases, according to the prescripts of the conference of bishops and without prejudice to 767.1
767.1. Among the forms of preaching, the homily, which is part of the liturgy itself and is reserved to a priest or deacon, is preeminent; (....)
So, non-ordained faithful can preach, but NOT in the Mass. Because homily in the mass is a part of the liturgy which is reserved to the cleric (priest and deacon) alone. The reason for this is answered in Instruction On Certain Questions Regarding The Collaboration Of The Non-Ordained Faithful In The Sacred Ministry Of Priest Article 3.1
The homily, therefore, during the celebration of the Holy Eucharist, must be reserved to the sacred minister, Priest or Deacon(69) to the exclusion of the non-ordained faithful.... This exclusion of the lay people is not based on their preaching ability of sacred ministers nor their theological preparation, but on that function which is reserved to them in virtue of having received the Sacrament of Holy Orders. For the same reason the diocesan Bishop cannot validly dispense from the canonical norm since this is not merely a disciplinary law but one which touches upon the closely connected functions of teaching and sanctifying.
PS: 'The homily should ordinarily be given by the priest celebrant himself. He may entrust it to a concelebrating priest or occasionally, according to circumstances, to the deacon, but never to a lay person. In particular cases and for a just cause, the homily may even be given by a bishop or a priest who is present at the celebration but cannot concelebrate.' (GIRM 66)
There seems to be a fine line though in what is an homily and another form of reflection. For instance, this article states:
As was the case with my own “reflection” at Sunday Mass (technically, a layperson cannot give a “homily”), many pastors do currently allow people other than priests and deacons to speak at Mass. It may be directly about the readings for that Mass, or it could be on a different topic that is relevant to the parish community. Occasionally it is just a talk by a member of the parish finance committee. [...]
In such cases the celebrating priest often gives a short homily or just makes a few comments before turning it over to the layperson. Pastors have mentioned to me that they often have to fend off criticism from a few folks for allowing laypeople to speak at Mass, but they make these exceptions in cases where there is an important message best delivered by someone with an expertise. [...]
It is suggested that laypersons may speak at other types of events, outside of Mass. In certain circumstances, they can speak during Mass, but this should never be confused with a homily.
Another gray area is that regarding the reading of the homily. Although the sermon can certainly be produced by the priest, it seems it could well be read by a layperson. This might well open the door to homilies which are prepared my priests in communion with laypersons.