The Leeds schools minute book states that on January 15th 1904 Mr T V Harrison, on behalf of the Hunslet Cricket, Football and Athletic Club presented to the schools union the handsome "Goldthorpe Trophy". In making the presentation Mr Harrison conveyed to the union the best wishes of the Hunslet Committee in their endeavours to promote schools football. The cup competition was run by the Hunslet Schools' RL from 1921 until 1996 when the association merged with the Leeds Schools Rugby League.
The article below, writtten by Mr J R H Greaves, and included in the 1952-53 handbook confirms that a knock-out competition was played in 1902-03 and it seems that, retrospectively at least, it was called the Goldthorpe Cup.
Arrangements for the 1904 final.
There was a dispute about ownership of the Goldthorpe Cup when Hunslet formed their own school association in 1921. Leeds claimed it should belong to them as it was presented to the Leeds Association, Hunslet claimed it should belong to them as it had been presented by the Hunslet club. In the end Leeds conceded and in 1924-25 Leeds schools withdrew from the competition. So from 1925-26 until 1972-73 (inclusive) only Hunslet schools took part in the competition.
Special thanks to Stephen Boothroyd for providing much of the information on the Goldthorpe Cup pages and also to Peter Todd for his help.