"Elles lisent nos lettre" is not correct.
If you want to use the singular for "lettre" then you also have to change the possessive pronoun : "Elles lisent notre lettre."
And there is a sound difference between this sentence and the one in the exercise. The difference lies between "nos" and "notre".
You can use this link to get used to the difference :
I hear nos not nous. It may be just a question of long, slow and laborious acclimatisation. People have the same problem with English. Different regions, different pronunciations. For example, in southern England, among speakers of Received Pronunciation, the words are, our and aah sound identical. I bet they are all quite distinct in American English. They aah reading aah letters.