Henry William Monckton was the eldest son of William Monckton-Arundell, 2nd Viscount Galway. He was educated at Eton between 1760 and 1765.
Henry took the additional surname of Arundell from 1769, following his father's inheritance of certain Arundell estates. He succeeded his father as Viscount Galway in 1772.
He served as M.P. for Pontefract 1772-74, but was not a prominent parliamentarian, making no speeches in the House of Commons. He died only eighteen months after inheriting the title, and was unmarried.
He was succeeded by his brother Robert Monckton-Arundell, 4th Viscount Galway.
He was unmarried.
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