When we use "have a" ...?
"Have a taste" is a bit more BrE, although some American speakers will use it. However, it's more common in AmE to say "Take a little taste" or "Try a taste". (Which is then usually followed by "Oh, c'mon, just a little taste....it's really good, you have to try it....." by some folks who refuse to take 'no' for an answer when it comes to sharing food.)
(Speaking of tasting and sharing food, my own Mother used to let Sparky the cat lick her ice cream cone, but she wouldn't eat off of the same fork as me if I offered her a taste of my food. Me, her own first-born child. :crazyeye: )