The idea of dropping sounds which we see in the spelling is always a bit strange to us, Spanish speakers of English. We’ve grown up thinking that every letter included in a word must be pronounced because this is the nature of our native tongue. But things are rather different in English.
At the most basic level, there are sounds that aren’t produced within individual words, the l in half or the s in island, for instance. We normally learn this very soon. But there are other sounds which are currently dropped in English as a result of connected speech processes, and this is a key factor if we want to sound more natural and fluent.
One of these cases is the dropping of the sound /h/ in the pronouns he, her, his, him. This is what native speakers normally do unless the pronoun comes at the beginning of a clause. So, it is not obligatory but it will certainly improve your fluency and will make your life a bit easier.
Today I have two good examples to illustrate this process. I find it very interesting that the speakers come from completely different backgrounds. The first one is a presenter from the BBC, the second is film director Elia Kazan in an interview from many years ago.
By the way, Elia Kazan is speaking about James Dean, if you’re interested.