Sutton was ordained a deacon and became chaplain to his cousin
Robert Sutton, 2nd Baron Lexinton,
English Envoy in Vienna in 1694. In 1697, he was appointed as secretary to the British legation there, and upon the departure of his cousin, became the English resident there. Lexinton then secured for him the nomination for
English ambassador to the
Ottoman Empire in
Constantinople on 5 December 1700, and he arrived in
Adrianople on 7 January 1702. Sutton asked to be recalled on 6 May 1715. He remained there until the summer of 1717, when he travelled to Vienna, arriving on 17 September. Afterwards, he served with
Abraham Stanyan as joint mediator at the Austro-Turkish peace congress at Passarowitz in 1718. His final diplomatic posting was as
ambassador to France in 1720, but was superseded the following year. Following his return to England, he bought estates in
Lincolnshire and
Nottinghamshire, worth nearly £5,000 a year, with a house at
Broughton, Lincolnshire. In Constantinople in 1704, Sutton acquired the
Arabian grey horse
Alcock's Arabian with some other Arabians, and had him shipped to England. The horse is considered to be the ancestor of all grey
Thoroughbreds. ==Politician and financier==