He then closed the Collegio Romano, on the plea of its insolvency, seized the houses at Frascati and Tivoli, and broke up the establishments in Bologna and the Legations.
In 1759 he was consecrated archbishop of Corinth in partibus, and in 1761 bishop of Frascati (the ancient Tusculum) in the Alban Hills near Rome.
In February 1798, at the approach of the invading French forces, Henry was forced to fly from Frascati to Naples, whence at the close of the same year he sailed to Messina.
He was buried in his brother's cathedral church at Frascati, but in 1807 his remains were removed to the Grotte Vaticane of St Peter's.
He held in succession the archbishopric of Siponto and the bishoprics of Sabina and Frascati.