From what we can tell from a variety of historical sources and registers, Humphrey White was the son of Thomas White (born 1677 - 1748). Thomas White had been an apprentice of Thomas Tompion, who is considered the father of English clock-making (famous amongst other things for creating the two very long pendulum clocks for the royal observatory for Charles II, which only required winding once a year and were accurate enough for performing astronomical calculations).
Humphrey must have started working in the family business as he grew up, taking over it before 1740 (8 years before his father passed away). The Whites, along with another clock-maker Richard Honeybone, provided clocks to most of the Cotswold area. Fairford clocks are still very sought after today in the Cotswolds. There is an auction of one of Thomas White's clocks this month. How wonderful it would be to reunite father and son's work after all this time!