Looking back, I would tend to agree with Osprey's comment (post 10). There could be some connection between the Emily Powell at Llanfrynach maried name Havard, and whose son possibly looks to have married Louisa Shayler after 1939. Cynog is John Cynog Havard's middle name, and Cynog could well be a reference to Merthyr Cynog - Upper Chapel (seemed to be a thing for people to utilise the place of birth as a name).
There are a couple of illegitimate births in the Builth area which are worth considering, as well as a couple of baptisms. However none quite match the birth quarter, but could have resulted in a registration in the following quarter. Baptisms could have been some time after birth and of course the supposed year of birth may still not be correct as far as birth registration goes - if in act the birth was actually registered.
There is also the possibility that the name was not necessarily Elizabeth Ann, but some variation or even slightly different name. I had a few in my tree who gave a different middle name to their birth certificate name - and of course all information passed down ultimately can be affected by memory.
One possible was
Elizabeth ann illegit dau of Eliza a single woman of the Drover's Inn Builth baptised at Aberedw may 7 1893.
Her mother is Eliza Powell servant at the Drovers Arms High st, Builth and is on the 1891 Census born 1865 at Upper Chapel
But the GRO has her birth as POWELL, ELIZABETH ANNE NO Mother's maiden name
1893 M Quarter in THE BUILTH UNION Volume 11B Page 69
The closest matches might be
POWELL, ELIZA ANN JONES
GRO Reference: 1892 S Quarter in THE BUILTH UNION Volume 11B Page 80
POWELL, LIZZIE PRICE
GRO Reference: 1894 S Quarter in BRECKNOCK Volume 11B Page 78
Besides the Eliza (mother) in the baptism above, there were a couple of possible Powell families in Upper Chapel that had daughters who could be candidates or a possible mother. There were a couple of non- Powell surname families who had Powell nieces and or nephews with them at some point and also likewise in Llanfrynach , where there were also Powell families (maybe all related). Some of these were however non-conformists, so records not necessarily readily available and a couple of them had 10 children, so not a quick job to eliminate all female candidates.
Any other random family clues that would help ?