Canon 31 was on the sons of parish canons:
In order to abolish a very bad practice that has grown up in many churches, we strictly forbid the sons of canons, especially if they are illegitimate, to become canons in the secular churches in which their fathers hold office. If the contrary is attempted, we declare it to be invalid. Those who attempt to make such persons canons are to be suspended from their benefices.
Illegitimate children were not the stigma that they have become in modern times (although it seems that having children without marriage is becoming more common in the 21st century, at least in the USA). A father might acknowledge an illegitimate child because it was a son and all he was having with his wife were daughters. The father would care for them and try to find careers for them. We saw this in noble families all the time. See here and here for examples.
In The Manly Priest, author Jennifer Thibodeaux discusses the difficult attempts to impose clerical celibacy during the period 1066 to 1300. Archbishop Anselm tried twice to prevent priests from having sex with women. One attempt was simply to say priests could not have wives living with them. The priests simply moved their wives to a different house.
Innocent's attempt at the Lateran Council was about as effective as Gregory's. His attempts at establishing clerical guidelines regarding dress and morality was one thing, but getting them to give up sex completely was doomed to failure. Thibodeaux calls this "an all-inclusive attempt to formally monasticize the priesthood and to separate them from the laity, while elevating them in status." [link]
In the canons of the Fourth Lateran, number 31 about sons follows number 30, which states:
It is very serious and absurd that prelates of churches, when they can promote suitable men to ecclesiastical benefices, are not afraid to choose unworthy men who lack both learning and honesty of behavior and who follow the urgings of the flesh rather than the judgment of reason. Nobody of a sound mind is ignorant of how much damage to churches arises from this.
It is clear that he two are connected because of the priest's desire to find a career for his son and the only career that the priest knows that he could educate his son about is managing a parish. The opportunity to place one's son in a position for which he was not suited was to great a potential problem.
Tomorrow we'll take a more detailed look at the idea of clerical celibacy and its origins.