Kate ate . . . . . . . . meat with . . . . . . . . vegetables for . . . . . . . . second course.

In the first blank, "no article" or "zero article" will be used because zero article is used 'Before the name of meals'.
For instance: Dinner, lunch and here meat is a meal. Therefore, no article will be used.
Similarly, in the second blank, 'no article' will be used because no article is used before the names of vegetables, plants etc.
In the third blank, article "the" will be used because article "the" is used to refer to a Definite person or a thing. It is also used before ordinals such as "first", "second" etc. Here, 'second course' is definite and specifically mentioned.