The direct object particle を stands next to the word of phrase that is the direct object in your sentence. This phrase in your translation is 「わたし の てがみ の うえ の つくえ」.
To analyse the meaning of this phrase, let's look at its parts:
「わたし の てがみ」 - My letter.
「わたし の てがみ の うえ」 - Above my letter.
「わたし の てがみ の うえ の つくえ」 - The desk above my letter.
So the direct object phrase is then "the desk above my letter" and your sentence is "Don't read the desk above my letter".
See how を stands next to the direct object "desk" and the rest of the phrase is used to further describe the desk. In your sentence, を stands next to the letter and makes the letter the direct object while the other part of the phrase describes the letter. So the book's translation sentence is "Don't read my letter on the desk" or in better English "Don't read my letter which is on the desk".