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This looks to be the 1841 census entry for the family
Gabalva hamlet, Llandaff, Glamorgan HO107/1418/2 folio 32 pg 24
David Jenkins 30 shoemaker
Lidiah 30
Mary 7
John 3
David 2
/Elvira Walters 13
John Smith 20 ag lab
all except Lidiah & Elvira born in county
1841 doesn't give places of birth, only county, or family relationships and ages over 15 are meant to be rounded down to the nearest 5 years.
I haven't spotted them as yet in 1851. Had they already left by then?
Most likely birth registration for Henry
Henry Jenkins june qtr 1842 Cardiff vol 26 pg 355
A copy of this birth certificate would give you his mother's maiden name and his exact date of birth.
I think the Elvira in the household is a clue as there's a possible marriage in St Mellons 15/5/1833 of a David Jenkins of Pentyrch and a Lydia Walters.
IGI extracted batch C023991 for St Mellons has baptisms for
Lydia Walter bp 14/4/1811 dau of John & Mary
Elvira Walter bp 25/1/1828 same parents
There are more siblings in the same batch.