"You read them a book."
Translation:Τους διαβάζεις ένα βιβλίο.
The problem with "διαβάζεις τους ένα βιβλίο" is the word order indeed. When you use the weak (shortened) form of the pronoun, it should precede the verb, otherwise you can use the strong type with a preposition "Τους διαβάζεις ένα βιβλίο" or "Διαβάζεις σε αυτούς ένα βιβλίο", the second sentence doesn't sound very natural though. "Διάβασέ τους ένα βιβλίο" in this word order can only be imperative ("διάβασε" alone could also be the third singular person of the simple past, but that needs another word order) so it shouldn't be given as a correct answer, because in English that would be "Read them a book"
Indication of the used pronoun:
"Τους διάβασες ένα βιβλίο" should be accepted since the verb "read" is spelled the same present/past
Ένα βιβλίο τους is one of their books. Τούς is the weak form of αυτούς and when is used for an indirect object, it comes before the verb. If you want you can say "Διαβάζεις ένα βιβλίο σε αυτούς=You read a book to them" So its either "Weak Pronoun form+ verb" or "verb+σε+strong pronoun form" There are more detailed answers here: https://www.duolingo.com/comment/17690328
Η γυναίκα μου διαβάζει ένα βιβλίο=My wife reads a book, Η γυναίκα μού διαβάζει ένα βιβλίο=The woman reads me a book, Η γυναίκα μου μού (the accent is not necessary on this one) διαβάζει ένα βιβλίο=My wife reads me a book.
Ι struggle with this too - when should εσεἰσ be used at the beginning of the sentence - I keep getting this wrong on various examples
There are just too many translations for this sentence and that one had been left out - added now!