a chóta = his coat (cóta is lenited)
a cóta = her coat (no lenition)
When the noun starts with a consonant which cannot be lenited, e.g. léine, then you have to infer from the context.
Example: Chuir sí a léine uirthi. We know it's her shirt because of the pronoun sí.
Tá a chóta agam, tá a cóta agam. It's only when the noun cannot be lenited as in the earlier example that you have to rely on the context.
The speaker is actually saying "a hool agus a ool" although it is easy to think she is saying "a oov". Listen very carefully and you can hear the "l" sound.
Rae F, you're right. It now sounds totally different than it did for me earlier today. I had listened to it multiple times this morning and then later when Moloughl wrote back and it sounded like ee. But just checked it again and it sounds like oo like I would expect. Maybe someone's just pranking the yanks. Thanks.
is it the addition of the ''h '' that determines the gender of the owner
Yes, for words that begin with a vowel. Please read Liamog's comment at the beginning of this discussion.
That’s true for nouns beginning with lenitable consonants, e.g. a pióg agus a phióg (“her pie and his pie”). For nouns beginning with vowels, hers gets a prefixed H and his doesn’t. “Aspiration” is how some books refer to lenition; the prefixed H is not an example of lenition.