He was born in the parish on 24 September 1646, and educated at Bury grammar school under Dr. Thomas Stephens, author of notes on Statius.
He was admitted a pensioner of Emmanuel College, Cambridge on 5 September 1659, before the age of thirteen, under the tuition of Thomas Jackson.
He was also presented to the rectory of Farnham Royal, Buckinghamshire by its Lord Keeper Francis North, 1st Baron Guilford, a kinsman of his wife's, and (instituted on 14 May 1683).
Along with his preacher's place at Gray's Inn, Clagett held the lectureship of St Michael Bassishaw, to which he was elected about two years before his death; and he also served as chaplain in ordinary to James II.
Clagett took a leading part in the controversy carried on during the reign of James II respecting the points in dispute between Protestants and Catholics.