Translation:The woman who is going to the cinema with me is my wife.
Sure, I believe you, but your wife at home may not! (Its late, just trying to be funny! Sorry, in advance, if I failed.)
Response: "The woman going with me to the movie is my wife", and the Correct Solution is indicating only the word 'cinema' as being wrong. Why in this case is kino not also good for movie?
We accept the American idiomatic phrase 'to the movies' (I had to add it here, otherwise that would be the suggestion you got), but 'to the movie' suggests just one specific movie, and that's "na film", not "do kina".
Sometimes people joke that if you go with someone "do kina", it's a date, and if you go "na film", then you're just friends.
Movie is singular, local speech pattern again, drives me to distraction.
"The woman accompanying me to the cinema is my wife." would be more likely English...