The peerage and baronetage of the British Empire as at present existing, 43rd ed, 1874, pp. 402-03 has some collateral branches of the Earls of Milltown, including some descendants of Hon. Robert Leeson.
After following a few different leads I found some details about the Henry Leeson and his son. The Plantagenet Roll of the Blood Royal: Anne of Exeter volume, by Marquis de Ruvigny and Raineval, p. 642 has Henry Corbett Leeson, d. 29 Sept 1887 (fourth son of Joseph Leeson, son of Hon. John Leeson, son of Brice, 3rd Earl of Milltown, and Anne O'Reilly) who married _____ Robinson, and had Henry Saunders Leeson (? dead)
Ancestry.com has more detail, Henry Corbett Leeson, married Frances Augusta Robinson, at Agra, Bengal 11 Sept 1857 and their son Henry Saunders Leeson, was born 17 Dec 1858 and baptised 16 Jan 1859 at Agra.
The Ancestry.com also has in Victoria (Australia) Police Gazette of 27 June 1888, p. 199 under a Missing Friends heading asking for information about "Colonel William Credqure Mitchel of the Bengal retired list who left India for Melbourne about 1865 accompanied by Mrs H.C. Leeson and her son, Henry Saunders Leeson, then 7 years of age. The colonel is required for the purpose of obtaining from him an account of the death of H.S. Leeson, before mentioned, who, it is stated was accidentally shot in 1879 or 1880, when returning to Melbourne from Brisbane. The information is sought by H. Corbett Leeson, of Delhi, Punjab, India."
So it looks like the son may have died in Australia. Not sure about the dates though, Henry Corbett Leeson, according to the Anne of Exeter volume died in Sept 1887, but the entry in the Police Gazette is dated 22 June 1888, published 27 June 1888. Either the date he died is incorrect or it is actually H. Corbett Leeson's estate who is trying to ascertain if his son has died?
I tried searching Trove the Australian newspaper database for anything about Henry Saunders Leeson, but didn't find anything and he may have gone under the name of Mitchel or something else. I'll have another look later.