First child born 1838, 1826 birth for him does not sound likely!
Does anyone have an actual sighting for him with Elizabeth or one of his children that would confirm age?
http://www.rootschat.com/forum/index.php?topic=725562.0 - it looks like there was an Archibald White born 1817 in Berwick, sailor, but whether he was the man married (? or not) to Elizabeth Knox is another matter. He does seem the most likely option.
Ditto whether he is the same man who died at sea in 1859 - it is entirely possible that your Archibald jumped ship somewhere else and lived out his life in another country.
http://www.nationalarchives.gov.uk/help-with-your-research/research-guides/merchant-seaman-serving-1858-1917/http://www.crewlist.org.uk/data/archivecrewlists/12667 - 12667 is the official number for the Humphrey Nelson - it looks like ship lists 1857/8/9 are at The National Archives (Kew). Hopefully these would show an age/birthplace for the Archibald White on that ship.
http://www.workhouses.org.uk/BerwickUponTweed/BerwickBirths1837-73.shtml - some of the children's births are in the Workhouse including a Robert b. 1854 and a Robert b. 1860
http://www.berwickfriends.org.uk/wp-content/uploads/2015/04/GBR-43-minutes.pdf - a mention here of Elizabeth White's illegitimate child in 1854, possibly conceived while she was a prisoner...
http://www.berwickfriends.org.uk/wp-content/uploads/2017/02/GBR-39-minutes.pdf - in 1843, Jane White, mother in prison, father deserted them
http://www.berwickfriends.org.uk/wp-content/uploads/2013/05/Newsletter-2010-September.pdf - here, a very colourful 1848 newspaper article describes Elizabeth Knox or White, aged 31, deserted by her husband "five years ago".
So by around 1843 he was gone (perhaps about long enough to father Margaret 1844 - or not?)
https://www.britishnewspaperarchive.co.ukPossible sighting:
Saturday 30 March 1844, Kentish Independent - Archibald White, a seaman, one of two men brought up for theft.
