I'm not really sure about this expression i found many times but never on grammar books:
I don't want him coming
(Want + sb/sth + verb+ing)
Is it formal, informal and is it better to use it in everyday speech or writings?
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Sure, you can say I don't want him coming. I don't see any issues with that. Usually you would use it with a prepositional phrase.
I don't want him coming to my barbeque.
I don't want that alcoholic coming to the wedding.
Without the prepositional phrase, it'd be more common to say:
I don't want him to come.
But I don't see any issue with I don't want him coming in spoken English.