"I always have a cup of tea and cheese" is something a native speaker might say but "I always have some cheese and a cup of tea" is better. Look at this: I always have a cup of tea and milk. I always have a cup of tea and lemon. I always have a cup of tea and cheese.

"Cheese" is uncountable, so you need to use "and (some) cheese" or "and a piece/slice of cheese". You don't use "a". I'd actually reverse the order of the tea and cheese, otherwise the reader might think that you put cheese in your teacup! "I always have some cheese and a cup of tea for breakfast."