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They were allowed to be married in the early Church but some like the Medicis or Borgias had children but were not married.
From the references:
George L. Williams, Papal Genealogy: The Families And Descendants Of The Popes, page 74: “Clement now made Alessandro de Medici “his illegitimate son by a slave” into the first duke of Florence” , McFarland & Company, 1998, ISBN 0-7864-2071-5
From: children of Popes
https://en.wikipedia.org/wiki/Category:Illegitimate_children_of_popes
- Cesare Borgia
- Gioffre Borgia
- Giovanni Borgia (Infans Romanus)
- Giovanni Borgia, 2nd Duke of Gandia
- Lucrezia Borgia
- Pier Luigi de Borgia, 1st duke of Gandía
How my heart sings 