Here is what the History of Parliament Online has to say about John's family, which lacks the names of the daughters, or his son.
Family and Education
b. c.1685, 1st s. of John Barnard, citizen (glover) and merchant of London, of George Lane, St. Botolph’s by Sarah, da. of Robert Payne of Play Hatch, Sonning, Berks. educ.Wandsworth sch. for Quakers. m. 5 Oct. 1708, Jane, da. of John Godschall, Turkey merchant, sis. of Sir Robert Godschall, 1s. 2da. Kntd. 29 Sept. 1732.
https://www.historyofparliamentonline.org/volume/1715-1754/member/barnard-john-1685-1764
I located his Memoirs on Google. That states that "Sarah, the eldest married Henry Temple, the Banker. {son of Viscount Palmerston{p 29) This edition is Prefaced by his great-grandson Thomas Hankey who claims that the son died "early and unmarried." No wonder there is confusion! He says that the eldest daughter (unnamed) married Sir Thomas Hankey.; and the youngest, also unnamed, married Mr. Temple, a son of Viscount Palmerston. (Preface to 1885 edition, p iv)
Memoirs of the Late Sir John Barnard, Knt. and Alderman of the City of London: Together with Notes Furnished by Mr. Overall, Librarian to the City of London Guildhall Library