They absolutely did, it was just about a common occurrence. I will mention some cases from the Late Middle Ages in Spain, because they were rather famous.

My lord Cardinal Pedro González sired three bastard sons with two different women. These bastards were recognised by him, and he loved them to no end. Being the extremely rich and powerful man that he was (he was nicknamed "The third King of Spain, the other two being Isabella and Ferdinand), he managed to secure some fiefdoms for his sons, the legitimisation from the Queen, and he even had them recognised as legitimate sons by the Pope.

The pope in question was none other than Alexander VI, also known as Rodrigo de Borja (or Borgia). Alexander fathered around a dozen children with a number of mistresses, the most prominent of which was Vannozza Cattanei, mother of Cesare Borgia, Lucrezia Borgia, Giovanni Borgia, and Gioffrè Borgia. By no means was this unheard of in the Roman curia.

In the archdiocese of Santiago de Compostela there were three archbishops of the same family. Alfonso de Fonseca I, who was father to Alonso de Fonseca II, who was in turn father to Alonso de Fonseca III, the most famous of the lot. Of course, as the Church's rules prohibits a direct succession, when Alonso I died, an Apostolic admnistrator was appointed for a year. Then Alonso II took charge of the business. Same with Alonso III.

Luis de Osorio, Bishop of Burgos, was the father of bishop Antonio Osorio de Acuña, better known as bishop Acuña. This Acuña became famous when the insurrection of the Comuneros in Castile in 1519, having joined the rebellion. His brother, a true knight and honourable man took the King's side.

Alonso de Aragón, bastard son of king Ferdinand the Catholic, was archbishop of Zaragoza. During his tenure, he sired a number of bastards, two of whom would become archbishops of Zaragoza in their turn.

The number of ecclesiastical bastards is absolutely massive, and these ones are but a few from a specific place.