Me again,
This is very confusing indeed and I don't want to make things more obscure but ...

The Monmouth Street noted in the 1871 is right on Shadwell Docks which would make sense if the father was a Mariner.
If this is the right family, the only thing I could find in 1861 that seems likely is ... wonder if Ingram/Elgram/Engram is all used in variation?
1861
3 Duke Street, St George in the East
Francis Ingram 1854 St George, Middlesex, England Son
Martha Ingram 1837 Not Known, Middlesex, England Wife
Mary Ingram 1858 St George, Middlesex, England Daughter
William Ingram 1860 (think is says 6 weeks) St George, Middlesex, England Son
Martha is a Tailoress. Her age is very difficult to read! but she is married.
No Samuel

RG9/279
Folio: 62
Page: 40
This address is again right near the Shadwell Docks.
+++
There is this too ...
Birth
Mary Ann Ingram
Jun Qtr 1861
St George in the East
Vol 1c Page 456
(from FreeBMD)
Sorry this might be leading right up the garden path ...

Cheers
Jonathan