The elision of the phoneme /h/ just occurs in certain types of words but, since they are very common words, it becomes incredibly frequent. So, we’ll distinguish three cases in which /h/ undergoes elision. The first two are by far the most important ones.
1) The function words he, him, his, her.
I wanted to forget her.
I met him at the theatre.
Does he know he’s going to be fired?
Andrew couldn’t make up his mind.
2) The auxiliary verb have.
My father had always been something of a maverick.
What have you been doing?
He’s too weak to have lifted it.
3) The pronoun who.
Did you know the man who called you?
The /h/ is frequently dropped in these words unless it’s stressed syllable or it comes at the beginning of a sentences or clause. For example,
He wasn’t aware of it.
I didn’t ask you. I was asking him.
Has she called your mother? No, she hasn’t.
Who did you ask for help?
And now some examples from native speakers:
First, the words he, him, his, her:
So, there’s a slightly competitive edge to what he’s writing (Esther Eidinow, BBC4).
And so when Facebook approached her for a job… (Dan Milmo, Today in Focus, The Guardian).
The decision was the result of a campaign by a Californian woman called Dorothy Mulkey. I’m Adam Smith and I’ve been to California to meet her (Adam Smith, BBC World Service).
When I met him I asked how he felt knowing the co-writer and director Paul King created the character of Phoenix Buchanan with him in mind (Samira Ahmed, BBC4).
I was like a father to him or a big brother to him and looked after him. Every morning, I’d call him out and see what he – condition he was in, whether he had a miserable night or not (Elia Kazan, speaking about his relationship with James Dean, NPR).
Second, the auxiliary have:
I found out that my car was rattling at speed because the tyres needed balancing. My car had always done that and I thought it was because it was old (Cambridge Proficiency test).
His reputation at the time, by the time of his death, was really very high but this didn’t translate into the sort of income that he might like to have earned (John Guy, BBC4).
It was the stronger messages that he would have liked to have got across (Lisa Jardine, BBC4).
Third, the pronoun who:
ln the theatre, the pleasures are immediate. You know exactly how you’re doing it, when you’re doing it. And it’s you who decides, really, how that is each time the curtain goes up (Cambridge Proficiency test).
As I see it, whether it’s at the level of, you know, impeachment or not, the Democrats are like the boy who cried wolf on this (Megyn Kelly, PBS).