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I posted the possibility of David being born before the 1892 marriage of John Williams to Hannah Mary Lewis in 1893. The 1911 census posted lists Mum as Hannah Mary. It's in the Carmarthan RD as are Shadrach and David's births.

Carl is this what makes you say all the children were called Lewis Williams?
eg.
Name: Mother's Maiden Surname: Order:
WILLIAMS, SHADRACH LEWIS
GRO Reference: 1896 S Quarter in CARMARTHEN Volume 11A Page 1141

Shadrach was registered as Shadrach Williams and it's the Mother's maiden name that is Lewis. the same follows for the other births I posted.

It would seem you're missing the birth for Gladys and Gwen. Look for gaps between births of the children that show on census.

The 1911 says the marriage had produced 9 children of which 6 were still living. So 2 or 3 births to find (David might not have been theirs) who presumably were born and died between census.

At only 40, Hannah Mary could have had further children too anytime post the census date.

If we have the right family your GG Grandad was John Williams b c 1870 and GG Grandmother was Hannah Mary Lewis b c 1871/2/3 (From 1901/1911 census)

?? This could be Hannah Mary's birth registration

Name: Mother's Maiden Surname: Order:
LEWIS, HANNAH MARY REES Order
GRO Reference: 1873 S Quarter in PONTYPRIDD Volume 11A Page 387

1901 says born Treherbert and 1911 Pentre (on original not Pentred as Transcript) - They are about 3 or 4 miles apart in the Rhondda Valley an area covered by Pontypridd RD

??
Surname First name(s) District Vol Page
Marriages Mar 1871 (>99%)
Lewis David Pontypridd 11a 390 <<<<<
Lewis Edward Pontypridd 11a 390
REES Mary Pontypridd 11a 390 <<<<<
Thomas Jennett Pontypridd 11a 390

an Edward and Jennett Lewis are on the 1871 census in Ystradyfodwg

Just posting whilst hunting around, these next 2 posts are very ??

The marriage certificate of Hannah Mary to John should reveal their Father's names and occupations but
???? could this be the family. Note the Father's name here and her own first child was David. According to genuki.org Ystradyfodwg is the former name for the Rhondda

1881 England, Wales & Scotland Census Transcription
38, Caroline Street, Ystradyfodwg, Pontypridd, Glamorganshire, Wales

First name(s) Last name Relationship Marital status Sex Age Birth year Birth place Occupation
David Lewis Head Married Male 32 1849 Carmarthenshire, Wales Coal Miner
Mary Lewis Wife Married Female 30 1851 Pontardawe, Glamorganshire, Wales -
Hannah M Lewis Daughter Single Female 7 1874 Treherbert, Glamorganshire, Wales Scholar
Elizabeth Lewis Daughter Single Female 6 1875 Treherbert, Glamorganshire, Wales Scholar
David Lewis Son Single Male 2 1879 Swansea, Glamorganshire, Wales -
Jane Lewis Daughter Single Female 4 1877 Treherbert, Glamorganshire, Wales -

And maybe in 1891. Original says b Blaenrhondda which is part of the community of Treherbert. David Lewis is 1 month old and born LLanddarog which is in CArmarthan RD so probably the birth I posted earlier Q! 1891 with no mmn

1891 England, Wales & Scotland Census Transcription
Tycsnol, Cwrt, Llanddarog, Carmarthen, Carmarthenshire, Wales

First name(s) Last name Relationship Marital status Sex Age Birth year Birth place Occupation
Levi James Head Married Male 68 1823 Carmarthenshire, Wales General Labourer
Hannah James Wife Married Female 72 1819 Carmarthenshire, Wales -
Hannah Mary Lewis Granddaughter Single Female 19 1872 Glamorganshire, Wales Servant Girl
David Lewis Great Grandson - Male 0 1891 Carmarthenshire, Wales -