I suppose the best way to describe the Irish pronunciation is that there's a "clipped h" at the end of the word "mhaith" - rather than a simple fading "aawww".

It's a perfectly reasonable and appropriate phrase to use in the context of the song. It is not, however, what the Clancys sing FWIW and , in the absence of evidence, unlikely to be the source phrase.

Drifting slightly - there's a Dublin slang word "mot" or "mott" for a girll/woman which is sometimes claimed to be derived from the Gaelic "cailín maith" which means "good girl". In fact, it was once much more widespread and probably derives from a Dutch word for a woman who was anything but good!

Regards