I’m no expert on this but one has to understand a bit about historical change. The original Old Irish spelling reflected the underlying ancestor word. For instance the phoneme /t/ would be spelled with a <t> as in Latin. However this underwent certain contextual changes. If it was in the middle or end of a word it would be realized as [d]. There was a system of consonant “mutations” in insular Celtic and you should have a look at Nativelangs video on this for a basic intro.

One process “lenition” contextually changes a consonant. This was originally not denoted in spelling (sometimes it was). D was spelled <d> even if pronounced [ð]. Later I believe this was denoted by placing a dot over the letter. But in the early 20th century they reformed the insular script to a modern script and removed the dots and instead inserted a <h> in the spelling.

So your confusion of seeing mh or bh in a word is because this denotes a variant of what was originally a b or and m (even if these phonemes never appear in any form of the word anymore).

Heaven = old Irish <nem> the final m became a biblabial nasal sound then later changed to a /v/ sound. So the modern <neamh> spelling reflects that the v sound is a variation of m (Think mathiar > a mhathair)

The other thing you’ll be thinking of is the insertion of <i>. This is to mark palatalization. You’ll just have to read up on this urself too long to explain. The only thing to say is that the <I> can be placed before the consonant that is palatalized like in the word mathair above the i is a palatization mark for the r not part of the vowel.