that is the word for word translation of the sentence. but in English that's just the way we say it. technically your answer isn't wrong, it's just not how it would be said in English, unless you're from some sort of old English movie haha. But to avoid saying the _ of _ , English speakers use the apostrophe with whatever or whoever the (owner) is. As in David's sister, rather than "the sister of David". Today's paper, rather than "the paper of today". I hope that helps.
I am not positive, but since diario translates to daily, the above example asks if you have the daily. Diardo de la mañana would be the morning daily.
Periódico means the periodical which seems to me becomes, idiomatically, newspaper. Indeed, both diario and periódico seem to be idiomatic. In English we sometimes use daily, but more often "today's newspaper".
Guess: both mean the same but could be used according to geography????
Superman, as Clark Kent wrote for the Daily Planet. :-)