During the projected Plantation of Connacht in the 1630s, his influence on Catholic members of the 1634 Parliament was so great that Lord Stafford issued warrants for his arrest, necessitating de Burgh hiding until the Viceroy's recall. On the recommendation of Bishop MacEgan of Elphin, he was appointed to Bishop of Clonfert on 16 October 1641, and consecrated on 9 April 1642. He subscribed to the ordinances agreed upon for the war against Parliament, spending almost all of his time in Kilkenny assisting the seventy-two-year-old David Rothe. During these years, Hugh de Burgh was agent for the Supreme Council at the Court of the Netherlands.
Upon the death of Malachias O'Queely, he was appointed Archbishop of Tuam by the Supreme Council of the Catholic Confederation of Ireland on 11 March 1647. On the same date he became archbishop, his brother, Hugh, was appointed Bishop of Kilmacduagh. He was noted as a good administrator of his dioceses, reforming many deep abuses, had churches repaired and established schools. In matters of policy, he was akin to his kinsman, Ulick Burke, 1st Marquess of Clanricarde. He became known for founding an extensive library in Tuam, which is thought was used by the likes of John Lynch and Dubhaltach Mac Fhirbhisigh. He also advanced a large sum to the Jesuits to maintain a seminary in Galway, in which town he built himself a residence three stories high.
He came into conflict with Walter Lynch, Vicar-capitular of Tuam, who was the Papal Nuncio's choice for the vacant see of Clonfert; however, de Burgh wanted it for his brother, Hugh. Giovanni Battista Rinuccini described the brothers as "hot-headed, and wishing to have everything their own way ... that it would be unwise to have two brothers collated to the two best dioceses in the province ... that the Archbishop of Tuam was the most unmanageable and refractory of all the Irish prelates with whom he had to deal. He blames me for recommending Lynch, and what is worse, he blames another who is superior to us all."
Fryde, E. B.; Greenway, D. E.; Porter, S.; Roy, I. (1986). Handbook of British Chronology (3rd ed.). Cambridge: Cambridge University Press. pp. 419 and 443. ISBN0-521-56350-X.