Is it Cúige Mumhan or Mhumhan? Since it’s Cúige Chonnacht, I’d have to think it would be Mhumhan, and I have found it online, but Mumhan seems to be much more common.

It's Cúige Mumhan, I'm 100% sure.
But after other nouns, one mainly uses "na Mumhan" = Gaelainn na Mumhan...

_________________
Is fearr Gaeilg na Gaeltaċta ná Gaeilg ar biṫ eile
Agus is í Gaeilg Ġaoṫ Doḃair is binne

OK, thanks.

I second that - if you look it up on the Irish language Wikipedia (Vicipéid) - it's on the Cúigí na hÉireann page.

Yeah, I assumed cúige was feminine after seeing Cúige Chonnacht, but it obviously isn't, so it's actually Chonnacht that's irregular.

The genitive of proper nouns is aspirated. Maybe they look at "Connacht" as a noun and the others as adjectives. (And if you buy that explanation . . . )

The names of the provinces of Ireland are, actually, the names of old tribes. So their nominative form is actually a nominative plural form, and their genitive is a genitive plural:
Connachta (= the people of Connaught) > Connacht
Mumha (=The people of Munster) > Mumhan
Laighin (=The people of Leinster) > Laighean
Ulaidh (= The people of Ulster) > Uladh.

Also, in the dative case, they may have an -ibh ending because they are dative plurals: in Ultaibh, go Laighnibh...

_________________
Is fearr Gaeilg na Gaeltaċta ná Gaeilg ar biṫ eile
Agus is í Gaeilg Ġaoṫ Doḃair is binne

except for Mumu/Mumhain. It is singular.

I guess Mumhan is just irregular in this case because it’s unlenited in the genitive?

There are quite a few strange usages in Munster Irish. There may be an explanation or explanations but I never get any satisfactory (i.e. grammatical/semantical) answers.