let $\lambda_{1},\lambda_2,\ldots\lambda_N$ be the eigenvalues of $M^{-1}XX^{T}$; the quantity you seek is
$${\rm E}[{\rm Tr}(XX^{T}\gamma/M)+I)^{-1}]=N\int d\lambda \rho(\lambda)(\lambda\gamma+1)^{-1}$$
where $\rho(\lambda)=E[N^{-1}\sum_n\delta(\lambda-\lambda_n)]$ is the eigenvalue density of Wishart matrices; this quantity is known in closed form for large $N$ (Marchenko-Pastur distribution).