It is likely that the infant John Bunham mentioned in Surgeon Bowes' journal (and in no other official documents) was James Baughan, the son of John Baughan and Mary Cleaver. He was born during the voyage, baptised after landing at Port Jackson on 27 March 1788 and died the next day.
* information from Mollie Gillen, The Founders of Australia: A Biographical Dictionary of the First Fleet (1989), pp 27-28
'Baughan, James (c. 1787–1788)', People Australia, National Centre of Biography, Australian National University, https://peopleaustralia.anu.edu.au/biography/baughan-james-30200/text37480, accessed 15 August 2022.
27 March,
1788
(aged ~ 1)
Sydney, New South Wales, Australia