No, I can't explain why your book says this is wrong. "I haven't been to the cinema yet." seems perfectly okay to me.
In fact all three options are possible...but they all have slightly different meanings.
Do they give you any other information about the situation which might explain why they think 'yet' is the wrong choice?
Here is a link explaining the rules for using 'yet' https://learnenglish.britishcouncil.org/intermediate-grammar/just-yet-still-and-already