I've had another look for Mary and think I have a possible family for her. 1851 census shows a helpful sister, although they both just have Wales as place of birth. Do both William & Mary die between 1861 & 1871? There's a possible for Henry Thomas from the 1861 census in an orphanage in 1871.
1851 73 Dublin St, Liverpool HO107/2177 folio 367 pg 76
Mary Harris head mar 32 husband at Sea b. Wales
Wm. Jas. son 3 b. Lancashire, Liverpool
Cathe. Cordon sister widow 30 b. Wales
Sarah Edwards servant 21 b. Wales
possible marriage
Catharine Morgan jun qtr 1844 Liverpool vol 20 pg 214 with Joseph Simpson Cordin on the same page
possible 1841 Lower Perry St, Liverpool HO107/568/24 folio 21 pg 1
William Morgan 45 boarding master
Mary 50
Lewis 15
Mary 20
Catherine 20
Ann 15
all not born in county
possible for sister Ann & mother Mary in 1851
3 Salthouse Lane, Liverpool HO107/2181 folio 125 pg 23
Ann Morgan head 26 victualler b. Wales, Llandilo, Carmarthenshire
Mary Morgan visitor widow 71 b. Llandilo
Thomas Jones 24 mariner b. Wales, Port Talbot
Harriet Guard 17 general servant b. Falmouth
There are possible baptisms for a similar family in Llandeilo Fawr, Carmarthenshire
baptism for Mary 12 Oct 1819
https://familysearch.org/pal:/MM9.1.1/X1RN-QLRand in same batch there are baptisms to the same parents for
Lewis 25/6/1826
Ann bp 22/12/1822
Catherine bp 11/5/1821
Margaret bp 10/7/1814
Thomas bp 21/7/1813
problem being that William does not show as a glover on the baptisms I have access to, but he is a victualler on Margaret's. Also, I'm reliant on a transcription for the marriage details, so the glover might not be correct. If you'd believe the transcription of the 1851 census, Mary's husband was at Tea!