Bedingfield was the son of Edmund Bedingfield of St. Clement Dane's, London, gentleman, and he matriculated from Hart Hall, Oxford on 2 April, 1737, aged 17. His degrees were B.A. 6 Feb., 1740-41, and M.A. 9 July, 1743. As "commensalis" of that hall he contributed a set of Latin verses on the death of Queen Caroline (1738). Some Latin lines by him in the Carmina Quadragesimalia (Oxford, 1748), ii. 14, are reproduced and much praised by Joseph Warton in his Essay on the Genius of Pope (1806 ed.), i. 47. The manuscript references in the second volume of the 1757 ed. of the Carmina which is in the Library of the British Museum assign to him several other pieces in the collection. The Education of Achilles was praised in the same essay of Warton (ii. 35) as a happy imitation of the manner of Spenser.
For some years Bedingfield seems to have lived in seclusion in his rooms at Hart Hall, or as it became later, Hertford College. Joseph Warton wrote to his brother on 7 June, 1753: "Give my compliments to Bedingfield. I am glad he is emerging into life from Hertford College." Some further poetical contributions from his pen were probably desired by Dodsley for the 1755 volume of the series, but in vain; for on Michaelmass Day in 1754 he writes to Warton: "I prodigiously admire your character of Mr. Bedingfield, who you say has actually refined his taste to a degree that makes him dissatisfy'd with almost every composition." A long letter on literary topics was sent by Bedingfield to Tom Warton on 3 March, 1757 (Wooll, Life of J. Warton, pp. 217, 225, 244).
Bedingfield was the acquaintance of a still more distinguished poet. He called on Gray in January, 1756, and left "a note with a very civil compliment." In October, 1757, he wrote to Gray "in a golden shower of Panegyrick." From 1757 to 1763 he seems to have lived at York, and then to have joined his family in Norfolk. Gray's letter to Mason [July], 1763, apparently implies that Bedingfield, like most of that race, was then a Roman Catholic (Letters, ed. Mitford, pp. 308-09, 313).
Edward Moore, the editor of The World, soon after the conclusion of that paper projected another magazine. Bedingfield was to have been a contributor, but the scheme dropped through Moore's death (English Poets, ed. Chalmers, xviii. 77).
Bedingfield is mentioned in a letter from Mason to Horace Walpole, 9 Aug. 1774 (Letters, ed. Cunningham, vi. 98), as then at York. This conflicts with the statement in Bell's Classical Arrangement (p. 21) that he died in 1768. His poem The Education of Achilles is reprinted in that collection.