In this paper, we study the existence and asymptotic behavior of solutions for a class of degenerate elliptic equation involving Grushin-type operator and Hardy potentials where $-(\Delta_{x}+|x|^{2\alpha}\Delta_{y}) $ is the Grushin-type operator, $\alpha>0, 2^*(s)=\frac{2(Q-s)}{Q-2} $ is the critical Sobolev-Hardy exponent and $Q=m+(\alpha+1)n $ is the homogenous dimension for Grushin operator. If $0 \leq \mu<(\frac{Q-2}{2})^{2}, 0 < s <2$, we will prove the existence of nontrivial, nonnegative solutions for this degenerate problem, and give the asymptotic behavior of solutions, at the singularity and at infinity.