Have breakfast is correct. It is one of the article use rules (or rather a set of exceptions) not to use the indefinite article with meals: breakfast, lunch, dinner, supper, brunch. Keep in mind that if you are adding an adjective to describe it, then you need to add one: I had a quick lunch, we are having an early breakfast, we had a delicious dinner. Of course, if you are referring to a specific meal, you need to use the definite article: I couldn't make it to the dinner, the breakfast they served wasn't great, the lunch went well.

no here you can say to have (cook, make, prepare) breakfast (lunch,tea,dinner,supper) without using "a" here but you have to use "a"when you want say to have a light breakfas, s small breakfast, a simple breakfast and so on

Have breakfast