On the moments of averages of quadratic twists of the Möbius function

We consider the moment of quadratic twists of the Möbius function of the form $$S_k(X,Y)=\sum _{d\le X}{(\sum _{n\le Y}\left(\frac{8d}{n}\right)\mu (n))}^k,$$ where $\left(\frac{8d}{\cdot }\right)$ is the Kronecker symbol and $d$ runs over positive, odd and square-free integers. We give unconditional results for their asymptotic behaviors.