Sometimes I hear people saying:
I'll take us some juice.
We were having us some juice.
I'll take me some juice.
And others
So, is there any specific use for that pronoun after the verb? Or it has the same meaning as the have/take alone.

Are you sure they speak standard American or British English? It sounds like it might be from a sort of dialect of English. Where I live nobody puts that pronoun there. The only meaning it might have, to my ear, is a sort of enthusiastic emphasis, but I can't say for sure because I don't belong to the group of people who use that construction.

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Hmm, so it certainly is a sort of southern dialect. Because I love country music [8]Emotion: big smile that unfortunately has a not much good grammar. But I like it, what can I do? Emotion: smile
At least if someday I go to southern US, I'll understand what people mean when say that.

I often use that type of construction. Where I live, we don't generally use 'me' - though I have heard that being used down in England - we use 'us' in both singular and plural cases. It doesn't really add any extra meaning to the sentence, but it is a very informal style.

Sorry then, I said it's bad grammar, but now I see it's cultural,I read somewhere that people in southern US use many constructions and words most used in british.
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There's no need to apologise. According to the Standard Form, it is bad grammar. Emotion: smile

I've noticed similarities between Scots English and the English used in the Southern States. But, that's probably because a lot of the settlers in the south were of Scottish descent.