I have found this family in 1851 census - think this could be John and his parents (aforementioned Michael and Ellen):
1851 Census - 62 Kent Street, St. George the Martyr, Southwark
Have been transcribed on Ancestry as Myens, looks like Megan to me.
Michael 35 Shoemaker (Master) born ? (looks like Yokyl) Ireland
Ellen 32 born Cork, Ireland
John 10 born London
Mary 8 ditto
Edward 7 ditto
Clara 5 (or 3) ditto
Catherine Singleton Sister 40 unm. Basketmaker
+ female assistant, 2 male lodgers, and 3 male servants (shoemakers).
ref. HO107 - 1563 - 169 - 6
If Catherine Singleton is Ellen's sister then this would support the 1840 marriage found, and also confirms they did indeed have a son John bc.1841.
What a nightmare this surname is.
I managed to use Catherine Singleton to get the census on FindMyPast and they have Mehigan as "Mesen" (no wonder I was having such problems finding them!) and Michael Mehigan's place of birth as "Yorkshire, England"!
I think that it looks more like "Megen" (if you look at the "g" in Singleton and the "a" in "Shoemaker" and "Master", the former has no loop on the "g" and the latter has quite an open loop which that fourth letter doesn't have -looks more like the closed "e" of the second letter).
I thought Michael's place of birth looked like "Yohyl, Ireland" so I Googled it and one of Google's suggestions is Youghal ("Older anglicisations of this name include Youghall, Yoghel and Yochil") which, perhaps coincidentally, is in County Cork too (although there is another town of the same name in Tipperary) and so unusually spelled that I can't really blame the enumerator for getting it wrong:
https://en.wikipedia.org/wiki/YoughalGiven the enumerator's flat "g" it could actually be "Yohigl" which is close enough for Google to offer Youghal as the right term (although this might be some wishful thinking and working backwards on my behalf):
https://www.google.co.uk/search?q=Yohigl+IrelandWorth noting that Catherine Singleton is dittoed in as being from London, which might suggest the Singletons were from London originally. This might fit with the family story that some ancestors on that side of the family went back and forth between England and Ireland (possibly as part of military service).