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Only one possibility at Aston cum Aughton:

John Handley, a farmer, bachelor, and Rebecca Darwent, spinster, both of this parish married after Banns 3 August 1780 in the presence of George Thomson and Benjamin Handley, Farrier? Furrier? or Farmer  Both signed the Register.

Added: the next entry in the Register is for John Darwent, a labourer, and Martha Duckenfield who were married at the same church on 29 August 1780.  John signed, Martha made her mark.

If you have just started researching, you’ve done really well to get back so far!  Amlwch is interesting.  Apart from being a port town it is renowned for the copper mines on Parys Mountain – no doubt where Richard learned his trade.  The mines date back to the Bronze Age, later the Romans.  The family of the Welsh painter, Kyffin Williams made their money there, by the time he was a young man most of it had gone but the extended family still owned a house in Venice where he produced some of his earlier works.

Forgot to say – marriage 2nd Nov 1833 at Llangyfelach, both Richard, a bachelor, and Hannah, a spinster, signed the Register by mark X, witnesses were Thomas Joseph and Thomas Charles.  Image on findmypast.

Richard Roberts baptisms - Amlwch
9 Nov 1811 parents John & Frances
2 March 1811 parents Richard & Anne
7 Aug 1813 parents Richard & Elizabeth

The names Frances and Elizabeth were not carried through by Richard and Hannah, so Richard and Anne seem the most likely parents for your Richard, but difficult to trace back further because there are several Richard Roberts / Anne marriages showing at Amlwch.

Searching An*ry for people by the name of Owen born Amlwch living in Glamorgan brings up several possible siblings for Hannah.  Perhaps they migrated there as a family?

Yes, those records came from findmypast.  I couldn’t see any others.

Looking at the birthplace for John 1828 I think Cwm is the property and Mynyddmawr the district.   There is such a place as Cwm showing on the NLS maps with the Llanelly & Mynyddmawr Railway running close by.  In the 1840s – 50s it was a 70 acre farm leased to the Gwendraeth Iron Company with Thomas Jenkins as  tenant in 1841.  Cwm is more or less centre of the map near the first L in LLANELLY, link here: https://maps.nls.uk/view/102179762

Following those census returns for David Jones b1828 you posted, I tend to think he is likely the same person as John b1828 – could be an error in the record or perhaps his parents simply changed their minds?  There are so many similarities; same occupation, same birth place and also David naming his first son Robert HENRY Jones [GRO 4Q 1851 folio 26, page 624 Llanelly.]  FindMyPast have this record showing his mother’s mn as Evans.  Remembering David was unmarried at the time of the 1851 census brings up a marriage 1851 2Q folio 26, page 950 Llanelly with David Jones and Elizabeth Evans on the same page.

I have left it too late to amend post #1 but ‘John’ was baptised 14 March 1828 not March 3.  The snip is from that record

According to the Calendar of Prisoners record available on Anc*y Annie was sentenced, 8 Feb 1889 to 10 days imprisonment followed by five years in a reformatory for stealing 8/- from a nine year old girl whilst the newspaper article attached below says 8d, which seems more likely  – can’t imagine a nine year old running loose with 8/- in her pocket in 1889!  Also attached is a previous article from 1888 giving some background to the 5 year sentence.

Looking for a girls’ reformatory in Lincoln brings up the Penitent Females’ Home.  http://www.childrenshomes.org.uk/LincolnPenitent/

In 1891 there were four staff and fourteen teenage inmates at the home, all listed as domestic servants but Annie is not one of them, so it does seem she was either let out early or ‘paroled’ for the remainder of her sentence.  Link to Anc*y record – 1891 Lincolnshire St Martin ED 31 pages 78 – 79. http://www.rootschat.com/links/01qxi/

Is this your Annie? 

added: An*y have her criminal record - two offences; the one below and five years 1889 - 1894 for stealing 8/- so still incarcerated in 1891