I have in my notes that mh makes the \v\ sound, and usually this seems to be the case.

Yet there are cases, such as, romham \r-oh-m\ where the mh seems to either dissappear altogether or perhaps serve as some sort of indicator. Is there a general rule I should be aware of? Or is the general rule that mh makes the \v\ sound except for particular special cases?

In Waterford Irish, at least, the usual rules (with exceptions) are as follows:

1. mh/bh at the start or end of a word = v.
2. mh/bh in the middle of the word immediately followed by a long vowel or a dipthong = v.
3. mh/bh in the middle of the word not immediately followed by a long vowel or a dipthong = silent.

Source: The Irish of Ring, County Waterford: A Phonetic Study by Risteard B. Breatnach §500-509.

Romham is one of the exceptions to the rule and is pronounced /ru:m/, in Waterford at least (§507).

I am only an early stage learner who just happened to have the book in front of me. Other people with far, far better Irish than me may be able to tell you which of these rules of thumb apply to other dialects and West Munster.

In school in Kerry we were taught to pronounce romham as /ro:m/ and romhat as /ro:t/, which I suspect is the more mainstream pronunciation.

Micheál Ó Siadhail in his book 'Modern Irish - Grammatical Structure and Dialectical Variation' gives the following default pronunciation for 'romhat': /rõ:t/, although he also has the letters 'Er' after the transcription, indicating Erris, Co. Mayo.

He goes on to say: "In Connemara, Tourmakeady, Ring and in Dunquin, the nasal causes the previous vowel to be raised e.g.romhat /ro:t/ -> /ru:t/"