This Mathew Gifford is not mentioned in Burke's Peerage. There is a cousin Mathew, son of a Bartholomew Giffard (of Clovelly-not Halsbury) that is mentioned, but that Mathew died without issue c1298, and he was not of Halsbury. The supposed father of this Mathew, Bartholomew Giffard who married Joan Halsbury is mentioned in Burke's Peerage and has an ancestry which in no way matches what I find for the Bartholomew, father of Mathew, in World Connect, raising the possibility that Mathew is not a son of the Bartholomew which married Joan Halsbury, but some other Bartholomew, with the ancestry given in World Connect.
Father: Bartholomew GIFFARD , of Halsbury b: AFT 1249 in Awliscombe, Honiton, Devon, England
Mother: Joan de HALSBURY b: ABT 1250 in Halsbury in Burrington, South Molton, Devon, England
Marriage 1 Spouse Unknown