From my research, I could only find one instance of one of the 36 Doctors of the Church who had a child. The Doctor is St. Augustine of Hippo, one of the 4 original doctors and he had a son named Adeodatus.
St. Augustine of Hippo, one of the most well-known and influential Church Fathers, had a son named Adeodatus, who was born to Augustine's partner before Augustine converted to Christianity. Adeodatus was born in 372 AD and died young, around the age of 17.
Augustine wrote about Adeodatus in his Confessions, which is an autobiography that covers Augustine's spiritual journey from his youth to his conversion to Christianity. In the book, Augustine talks about the love he had for his son and how he struggled to reconcile his love for Adeodatus with his desire to follow the celibate life of a Christian priest. He said that Adeodatus was a source of joy for him, and that his birth had been a sign of God's grace.
Adeodatus was highly intelligent and well-educated, and Augustine took great pains to provide him with the best education available. According to some accounts, Adeodatus may have been considered a prodigy, and Augustine mentions that his son was able to read and write at a very young age and was highly interested in philosophy.
Although Adeodatus died young, he made a lasting impression on Augustine, who wrote fondly of him in his Confessions, and considered him a reminder of the beauty of God's grace in his life.
Is anything known of Adeodatus's mother?
The identity of Adeodatus's mother is not entirely clear. Augustine never mentions her by name in his writings and what little is known about her comes from a few passing references in his works. According to his own writings in his Confessions, Augustine had a relationship with a woman, who is not named, before his conversion to Christianity and Adeodatus was born from this union. It is not clear whether the woman was a wife, a concubine or a partner in a informal union. But it is believed that she was not a Christian and Augustine did not live with her after his conversion.
Why aren't there more children of Doctors of the Church?
The title "Doctor of the Church" is a recognition given by the Catholic Church to certain saints who have made significant contributions to the development of theology and doctrine. Typically, these individuals have been priests, bishops, or monks who were actively involved in the Church and made important contributions to theology through their writings, teachings, or leadership.
There are 4 female doctors of the Church including St. Catherine of Siena, St. Teresa of Avila, St. Teresa of Lisieux, and St. Hildegard of Bingen. Obviously none of them were clergy. The two Teresas were nuns, St. Hildegard was an abbess, and St. Catherine was a lay person who was a member of the 3rd order Dominicans.
Other than the women, it seems all the Doctors of the Church have been clergy: priests, bishops, or popes.