Discrete time system signal flow graph problem
There is clearly recursion in there and your solution does not reflect it since its purely FIR.

I recommend doing this as a coupled difference equation: Define a new variable $v[n]$ at the point where Peter drew the circle and than derive the transfer function from (1) X->V and then (2) X and V -> Y. Then you can eliminate V in eq (2) by using equation (1).