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I've tried to find Newport Market records before, without much success. There's a useful article here ...
http://www.childrenshomes.org.uk/NewportMarket/The institution was absorbed by the Shaftesbury Society in 1928, so any historical records will be in that collection at London Metropolitan Archives (LMA/4647). But their online catalogue shows that the records they hold are nearly all administrative, apart from this group ...
Homes and School records
LMA/4647/D which includes records from most of the homes run by the Shaftesbury Society, although the records of individual admissions to the homes are very sparse.It doesn't look very hopeful, but you could email LMA to ask what records of individuals they have for the relevant period. Queries about their holdings are normally answered free of charge.
ask.lma (AT) cityoflondon.gov.uk