Terrick was educated at Clare College, Cambridge, graduating BA in 1729 (MA in 1733) and DD in 1747.

He was appointed Bishop of Peterborough in 1757 through the influence of the Duke of Devonshire, the then Prime Minister, but subsequently transferred his allegiance to the Earl of Bute.

He declined the archbishopric of York in 1776 on the grounds of ill health, dying on Easter Monday 1777.

Horace Walpole, who disliked Terrick, said he lacked ability, save "a sonorous delivery and an assiduity of backstairs address".

On the other hand, Alexander Carlyle thought him "a truly excellent man of a liberal mind and excellent good temper" and "a famous good preacher and the best reader of prayers I ever heard".