"Matěj prefers tea."
Translation:Matěj má raději čaj.
Rád is not a verb. It is either an adverb or an adjective (depending on the context).
In this sentence, it is a part of the predicate - "mít raději" and it is a comparative form of the verb "mít rád" = "to like".
There is also a superlative form - mít nejraději.
rád - raději - nejraději / mít rád - mít raději - mít nejraději
I'm not sure it means "rather have" in the sense of, right now, making a choice between having, say, coffee or tea. I think it means that what he "has" is a general preference for tea over, say, coffee. "Matěj would rather have tea" -- in the making-a-choice-now sense -- might be a different sentence.
But I use plenty of "in my head" pseudo-translations to help things make sense, so if your suggestion works for you, that's all good! And, besides, my theory might be wrong. :-)