Just come across this for the first time on another board, meaning ‘plead poverty’. All the hits I get seem to be American. Irritatingly, the first hit on Google if you put double quotation marks round it is a JSTOR site where the tiny extract that Google prints seems to hint it may be of Irish origin, but it’s difficult to tell, and the actual site is subscriber-only.

Are others familiar with the phrase?

This is originally more or less a Scotticism, I think. “Poor-mouth” (v.) is general US now, says DARE. Constructions such as “make a poor mouth” are more regional (Midland, Southern).

There is a transitive use which deviates in meaning, I suppose more recent, = “disparage"/"downplay"/"bad-mouth".

SND shows examples of “mak a puir mouth” etc. back to 1822, under “mouth”:

http://www.dsl.ac.uk/dsl/getent4.php?plen=26816&startset=26085849&query=MOUTH&fhit=puir&dregion=entry&dtext=snd#fhit

My father (US) used to use “poor-mouth” (v.i.) = “speak as if poor” ... also “talk poor-mouth” with “poor-mouth” acting like an adverb.

There is a transitive use which deviates in meaning, I suppose more recent, = “disparage"/"downplay"/"bad-mouth".

Yes, OED’s earliest cite for this sense is 1948.

I remember reading, and enjoying, “An Beal Bocht” or, in English (the book was originally written in Gaelic), “The Poor Mouth” by Flann O’Brien.  Either in the introduction or on the cover it was explained that “to put on the poor mouth” was to plead poverty and this was the origin of the title. This suggests that the saying was originally Irish Gaelic.

The existence of an equivalent phrase in Irish does not necessarily indicate Irish origin IMHO. Isn’t there an influence of English on Irish? If the expression was widely used in Scots [English] in the early 19th century, I suppose it would have appeared in [at least parts of] Ireland pretty quickly, whether a comparable Irish expression already existed or not.

Yes, but would it have been translated into Gaelic in that case (unless it originated in Scots (Gaelic)?

And is it not equally possible that the phrase also existed in Scots Gaelic, and was translated into English as Highlanders dispossessed by the Clearances moved into cities such as Glasgow at the end of the 18th century?