As I see, your topic is Latin infinitives. You have to choose between these verbs: legere, legisse, lecturos esse, lecturas esse, legi, lectos esse, lectas esse. I've put verbs from English and Latin versions in bold, to show how it translates.
1. Referring to the English translation: "I think that the boys read the books.", correctly completed sentence looks like this: Puto pueros libros legere. Legere = read or I/he/she/it am/is reading. Because it is present tense.
2. According to the English translation: "I think that they will read the letters.", the sentence should be completed like this: Puto eos litteras lecturos esse. As you see Lecturos esse translates like will read or be reading (future tense).
3. "I think that the girls are reading the books.", so the sentence should be completed like this: Puto puellas libros legere. As I've mentioned it before, in the cases of present simple it translates as are reading or read.
4. According to this translation: "I thought that the students were reading the books.", the sentence can be completed like this: Putavi discipulos libros legere. (Past tense).
5. The English version is: I thought that the soldiers had read the books. So the Latin version should be completed like this: Putavi milites libros legisse. (introductory perfect verb).
6. We know that the English version is: I think that the letters are being read by the boys. So the Latin version will be: Puto litteras a pueris legi. (are read or are being read.
7. If the English translation is: I think that the girls will read the books. The Latin one will be: Puto puellas libros lecturas esse. (Notice that this one is the same as the second sentence, but feminine).
8. We have English version of the sentence, that is: I think that the letters have been read by them. So the Latin one should be translated like this: Puto litteras a eis lectas esse.
9. Here we've got English translation:
I thought that the books had been read by the soldiers. The Latin version should be completed like this: Putavi libros a militibus lectos esse.
10. According to the English translation: "I thought that the books were being read by the soldiers.", the Latin version should be completed like this: Putavi libros a militibus legi.
