"Il a pu lire ce livre."
Translation:He was able to read this book.
Although I have seen situations where "Il a pu..." was translated as "he has been able...", my sense is that it is a colloquial use. Il a été pu lire ce livre = He has been able to read this book. The problem stems from the fact that Passé composé does not map directly to a single English verb tense. Passé composé can be translated as either English simple past, i.e. il a parlé = He spoke, or as English present perfect, i.e., il a parlé = He has spoken. Passé composé does not map to English present perfect continuous "He has been speaking". That would be "Il a été parler". Even so, the landscape of grammar in both French and English has many different advocates and usage varies considerably.
That is the pluperfect tense. Il avait pu lire ce livre. Note that the verb "pouvoir" contains the work "be" as in "to be able", so that it is appropriate to use the Passé composé "Il a pu" = he has been able and the pluperfect "il avait pu" = he had been able. This does not work with many French verbs, however, which may call for the use of être to make it work, e.g., Il avait été parler = He had been speaking. Compare to: Il a parlé = He spoke, He has spoken, He did speak. But not "He has been speaking". Aside: The mapping of French verb tenses to English verb tenses is not one-for-one and becomes especially problematic to speakers on both sides when continuous tenses are used because French does not use a specific continuous tense.
The problem is that "could" is used two different ways in English: 1) past tense of "is able to", and 2) present conditional meaning that the possibility of the action actually taking place is uncertain or not guaranteed to occur. So with the French "a pu", we can know that it is referring to his ability to do something in the past and not the conditional mood. It's true that "He could read that book" is ambiguous in English just like there are sometimes ambiguous sentences in French but it is nevertheless completely accurate.
I'd translate your second sentence as Il aurait pu lire ce livre, i.e., he had the chance to read it — and, incidentally, depending on the intonation, he had the chance, did not, and he should be blamed for that. As for He could read that book, I'd translate that as Il pouvait lire ce livre. HTH!
Thanks for your help :) Duolingo had told me that "He could read that book" was the right translation. Now, above, I can see where it says that "He has been able to read that book" is the translation, and THAT makes more sense to me. Before, I could only think of a scenario such as a Mom in a book store thinking of her child: "He couldn't read this book. He COULD read this book. I'll buy him this one."
I should tried to sleuth out why "She could sleep" was given as "Elle pouvait dormir" and "Elle pourrait dormir" (I think it was the infinitive, I may be mistaken). But given this example wouldn't "Elle a pu dormir" work just as well. I think I need to find a grammar book! Au secours!
Using your Mom in a book store example, I think she would more likely say "He can't read this book (it is too complicated for him) but he can read this one (it is simple enough)" if she was referring to his ability. The statement "He couldn't read this but he could read this one" seems to me something she would say referring to something in the past. When he was much younger, he couldn't read a book with small lettering and long words but he could read a book with big letters and short words.