Have you drunk water/coffee? 

Have you had water/coffee?

Which one is correct or more commonly used? Any more examples for such questions?

1 2
Neither is particularly natural. I suggest:

Have you had any coffee yet?
'Have you had any water yet?'

Is this fine as well?
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Yes, though I see little use for the phrase.
1. How do you normally ask a child if he has consumed water or not? 

2. Is there such a usage using 'have you drunk...'?
There is nothing grammatically wrong with your sentences; they just do not seem usual to me. For instance, I don't think I have ever asked a child whether s/he has drunk any water. Children start whining that they are thirsty long before we think they might be parched– which they seldom or never are anyway.
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It's interesting; I'm not sure between our cultural differences. We normally show concern (health) for the son and ask (in Mandarin) if the child he has drunk water yet?  It's pretty common here. Don't people in western countries practice it?
We are concerned about our children's health, but we do not generally need to know whether they have had any water-- they drink plenty of milk, fruit juice and other beverages in the course of the day.
Oh I see!
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