This is what I can discover about Philadelphia and Eliza from the Feiling biography. Feiling himself does not mention the possibility that Philadelphia had an affair with Hastings, but it was written in 1954 when people were more reticent (though it does talk about another married woman with whom Hastings had an affair, the Baroness Anna Maria Chapuset Imhoff). To summarise what Feiling writes:

Hastings made friends with the surgeon Tysoe Saul Hancock, and his wife Philadelphia, `who was born an Austen and aunt of the immortal Jane'. Hastings was godfather to their one child, Elizabeth. His son George lived in the care of `Philadelphia's brother George, father of Jane and vicar of Steventon in Hampshire, where he died in 1764, of what sounds like diphtheria', just before Hastings returned to England. Hastings and Hancock became partners in a number of trading ventures.

During 1765-9, Hastings was back in England not doing very much, and the Hancocks had travelled back to England with him. But in 1769 Hastings was given another Indian appointment. Hancock had already returned to India, but Philadelphia stayed in London; she continued there while both Hastings and Hancock were in India.

Hastings arranged to spend 20 pounds a year to a Mrs Forde for the care of his stepdaughters (not much, was it?); before he returned to India, Hastings also left `some money' for Philadelphia: how much, and whether it was to do with his partnership with Hancock or something else, Feiling does not say, curse him.

Later on, 1772-4, Hastings started to make more money. Again summarising Feiling: there was serious trouble with one of his stepdaughters who had been prepared for India by dancing lessons and other `genteel accomplishments'; she ran off and married a corporal (of whom no more is said, alas).

Hastings appears to have remembered his friends, with a pension of 250 pounds a year to Mrs Vernet (widow of a man who had saved his life), 350 pa to the Baroness Imhoff and her husband, and a bond on China, estimated to be worth 5000 pounds, to his god-daughter Elizabeth, which is a lot of money in comparison with what he was giving other people.

Hancock was aware of Hastings' affair with the Baroness Imhoff, and wrote to tell `Phila' in London about this `principal favourite among the ladies' ... who `has been very pretty, is sensible, lively, and wants only to be a greater mistress of the English language to prove she has a great share of wit'; her husband was `truly a German'. I would just love to know what Hancock was thinking when he wrote to tell his wife that Hastings had a new favourite among the ladies!

That's as far as I've read in Feiling: it's heavy going. I'll post more if I find it.