Sorry this is a reply to an old post but this exact question was doing my head in, because if (as ChatGPT wrongly told me) ‘a dweud y gwir’ means ‘and to tell the truth’, it could only really be properly used after some other statement on the general topic under discussion.
The online Gweiadur seems to have the answer (definition 5 below). The ‘a’ in ‘a dweud y gwir’ has no direct English translation, giving it a particle - like feel, however it also hints at a general meaning in the a/ac/â broad family of words not a million miles away from ‘and/with telling the truth.’ However it is a separately categorised use of the word distinct from a/ac/and’, neither is it ‘â’, having no to bach. Use 5 of ‘a’ is as a word (or particle?) specifically to introduce certain phrases such as ‘a dweud y gwir’, ‘a dweud y lleiaf’, ‘a bod yn honest’, ‘a thorri stori hir yn fyr’ (note aspirate mutation as with ‘and’). I’m not sure if this is grammatically sound (especially since it is listed as a conjunction), but I am going to think of it as a particle, because this way of thinking frees me from the frustrating question, ‘what does it mean’?
a2:ac
conjunction
1i gysylltu geiriau unigol, ci a chath, oren ac afal, ac ymadroddion, Mae un yn y fasged ac un yn y bag. Rwy’n mynd i’r dref a byddaf yno drwy’r dydd.
2ynghlwm wrth arddodiad i ddynodi meddiant neu berthynas, gwraig a chanddi lygaid mawr, bwthyn ac iddo ddrws glas
3mewn ymadroddion dyblyg nad ydynt yn enwi person, peth na lle, y dyn a’r dyn, y lle a’r lle
4wrth ysgrifennu symiau o arian, punt a hanner can ceiniog
5i gyflwyno ymadroddion megis a dweud y gwir, a thorri stori hir yn fyr, a dweud y lleiaf, a hithau wedi dechrau nosi
6i gyflwyno rhagenw mewn cyfosodiad, a minnau, brifathro ysgol
7i gyflwyno digwyddiadau sy’n gydamserol, A minnau’n sefyll wrth ochr y ffordd, gwelais y car coch yn taro’r car melyn.