Énri Ó Connmhaigh (fl. 1405) was Bishop of Clonfert and Bishop of Kilmacduagh. Bishop Ó Connmhaigh was one of at least two bearers of the surname (now rendered as Conway to hold this office. He was succeeded by a Seaán Ó Connmhaigh, who became bishop in 1441. An apparent kinsman, Máel Muire Ó Connaig, who held the office from may have been a relative who bore an earlier version of the surname.