I read this article and now I'm confused when got can be omitted when using have.
Could this be explained in plain English without technical terms?
Is there a different usage in past tense?
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You can safely omit it pretty much always.
I think it's largely a regional variety issue. As the article explains, in the UK "I've got" might be used more commonly where Americans would say "I have".
I remember being taught phrases like this in elementary English classes (due to British English emphasis in Finnish schools at that time):
I've got a cat.
But rest assured, it is never wrong to omit "got" in a phrase like this and just say:
I have a cat.
Also in those cases where it's used to add emphasis or indicate obligation...
I've got to go now!
... you can always omit it and still be perfectly understood (you can add the emphasis in other ways):
I have to go now!
We hear "I got it" and "I've got it" for "I have it." One day our teacher went down the roll list to ask each of us whether he had his homework. "Got it" was the nearly unanimous positive response—until the teacher reached me. "Have it," I announced, and the classroom burst into laughter. "I got it" means "I received it," not that I currently have it. "I've got it" sounds right through years of usage, but try to explain how "have" and "got" logically form an idea together. K.I.S.S.—keep it simple, stupid: "I have it."