It is pretty straightforward, though.

There is no lenition nor eclipsis after it, and it itself ends in a vowel, thus it prefixes
h- to vowels – that’s a common pattern. And generally when something causes the
h- prefix, then it
does not cause lenition or eclipsis.
For example
le behaves this way (
le Máire,
le Pól, but
le hÚna),
go meaning ‘to’ (
go Corcaigh but
go hÉirinn), or
chomh (even though in writing it has a final consonant, but pronounced /xo(ː)/,
chomh beag, chomh hard), to some extent the negative copula
ní (in the standard it’s limited to pronouns and some words, eg.
ní hé, ní hamháin, in older texts, proverbs, and some dialects you’ll see the
h- appearing before all kinds of stuff beginning in a vowel).
So it’s either lenition or
h- before vowels, never both. ;-)
(And the historical reason for this is that
most Irish words ending in a vowel that don’t cause lenition or eclipsis, in pre-Old Irish times used to end in
-s, this
-s got lenited to
h and dropped before consonants – but blocked lenition – and survived only before other vowels. The ones which originally didn’t have the
s, but did not cause lenition or eclipsis, started appending
h- by analogy to the other ones.)