Out of interest, have you located the birth of the child that had died by 1911? I can't seem to find one with mmn COTTON.
I had not noticed that they had another child to be honest. But I had looked on GRO going right back and nothing in any of the years for a Boxall with mother Cotton. Or Cotton as surname.
Just wondered, since we've got two missing births, apparently neither with maiden name Cotton. I doubt Charles will be under Boxall, since I don't think James John was his biological father (also notice on the 1911 where they incorrectly noted children on both James and Hannah, he says he had one child born and died, she says two children born- she's counted Charles where he hasn't, even though they've amended their years married to include his birth). It's possible the dead child was stillborn (which would be another error on the census, since they weren't supposed to count children not born alive), in which case there won't be a birth registration. If there is one though, it may well be under a different maiden name for some reason, since we can't find one under Cotton. I'm somewhat minded of my 2xgt grandparents, who registered their daughters correctly but I always joke my 2gt grandfather must have wet the baby's head a little too enthusiastically when his son was born because he put his wife's middle name as her maiden name, causing confusion in his descendants

The same couple... My 2xgt grandmother had an illegitimate child prior to the marriage and pretended to be married to the father on the child's birth certificate, which was registered at her mother's home address in Cornwall. 2xgt grandfather recorded on the census that the child was born in Folkestone, Kent. Where was the bio-father eventually found? Folkestone, Kent

I don't know if 2xgt grandfather got confused between where the child was born and where she was conceived, or if my 2xgt grandmother gave birth in Kent and did a naughty registering the child when she got home.