According to Francis K. Mason P.1120 was Hawker's submission for an advanced trainer (Specification T185D) which was won by Folland with the Gnat T.1. I expect there's something on it in the T185D papers at Kew:
Whoops, yes I would concede HAL as being Hawker Aircraft Limited.
I am currently doing fairly extensive research on RAF trainers (hence my interest being piqued here) and I have found no mention of the P.1120 (or any Hunter development) in connection with T.185D in the Kew files. By 1958 work was well underway on the Gnat trainer to fulfil T.185D and T.185 seems to have been written around the Gnat in response to studies undertaken by Folland in 1956 at the MoS's behest.
Maybe this was an early attempt to sell a T.66 to the RAF? By 1961 the RAF was certainly interested in the T.66 as a TSR.2 lead-in trainer.
One thing that seems to be recurring is that the info on project studies in the old Putnams is often showing up as incorrect or mystifying.