He was the eldest son, and heir apparent, of Henry Hamilton-Moore, 3rd Earl of Drogheda and Mary Cole (d. 1726).
His maternal grandparents were Sir John Cole, 1st Baronet and Elizabeth Chichester (a daughter of Hon.
[1] Lord Moore was a Member of the Irish House of Commons for Drogheda from 1692 to 1699 and, again, from 1703 to 1713.
[2] As the heir apparent to the earldom of Drogheda, he was styled Viscount Moore.
Moore predeceased his father, however, so the earldom passed to his eldest son, Henry, in 1714.