Different ways to pronounce "augh"
Solution 1:
The real bugger here is -ugh, plus a preceding vowel. The variants are:
- [ɔf] - cough
- [ӕf] - laugh
- [ӕu] - bough
- [u] - through
- [o] - though
- [ɔ] - bought, naught
The problem with all of these is the sound formerly pronounced as [x], which has disappeared in Modern English. The way that it disappeared, though, was variable. In most cases it simply vanished, mutating the vowel preceding it in various ways, with further vowel mutations influenced by a following consonant. This was what happened in through, though, bought, etc.
The unusual case is the words in which -ugh represents [f]. This is, alas, an irregular sound change. The ordinary outcome of the Old English form of laugh would be something pronounced like law, but for irregular and unpredictable reasons the [x] became [f] in this case.
Solution 2:
What about braugh in the phrase Erin go braugh, described by Wikipedia as an anglicisation of the Irish Éirinn go Brách? Does that work? It's pronounced as "bra", with a bit of a trill on the "r".
Solution 3:
From my dictionary (I also put the ones you mentioned):
- caught |kôt| - ORIGIN: Middle English (also in the sense [chase]): from Anglo-Norman French and Old Northern French cachier, variant of Old French chacier, based on Latin captare ‘try to catch,’ from capere ‘take.’
- naught |nôt| - ORIGIN: Old English nāwiht, -wuht, from nā [no] + wiht [thing] (see wight).
- laugh |laf| - ORIGIN: Old English hlæhhan, hliehhan, of Germanic origin; related to Dutch and German lachen, also to laughter.
- aught |ôt| - ORIGIN: Old English āwiht (see aye, wight).
- daughter |ˈdôtər; ˈdä-| - ORIGIN: Old English dohtor, of Germanic origin; related to Dutch dochter and German Tochter, from an Indo-European root shared by Greek thugatēr.
- faugh |fô| (DATED) - ORIGIN: natural exclamation: first recorded in English in the mid 16th cent.
I found more but they are more or less the same pronunciation, so I decided not to add them.