English speakers often use the phrase "have ___" instead of "drink ___". For example:

Do you want to have a glass of wine with me?

You use "have a drink" when you're talking about the experience of drinking something, or you're talking about drinking in a social situation.

If you're describing what you drank in a scientific- or medical-sounding way, you would use the verb "drink" instead of the phrase "have a drink". For example, if you got sick while on vacation in a foreign country, you might describe what made you sick:

I drank some fruit juice with ice in it on the second day, and I think that the ice is what gave me the stomach virus.