I realise this is not what you are seeking, but I have found them in Eastbourne in 1921, mother in Princess Alice hospital, two children alone in Marine Parade.
In 1916 she is advertising in the Kent and Sussex Courier to buy old clothes from 257 Seaside, Eastbourne. I checked that address in 1911, but she is not there.
I guess that you know about the 1916 newspaper report about "Matrimonial Difficulties"? From that article:
Mrs Eva Gausden deposed that she was married in...1914.. [and] lived with him [George Gausden] in Carlton-road until she went into business about a year ago.
This is his address in the 1911 census, but as you say, she is not there. This article also states:
She had kept the three children during the past month.
Was there another child that you know of?