He was the second son of General Henry Conyngham of Slane Castle and his wife Mary Williams, daughter of Sir John Williams, 2nd Baronet, and widow of Charles Petty, 1st Baron Shelburne.
He succeeded to the family estate on the death of his brother William.
An absentee landlord, he owned extensive properties in counties Meath and Donegal, while spending most of his time abroad.
Despite being a British MP, he also controlled the pocket Irish constituency of Newtown Limavady and sat for Killybegs between 1727 and 1753.
On his death all his titles became extinct except Baron Conyngham, which passed by special remainder to his nephew Francis Burton, who adopted the surname Conyngham.