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 $N,M\rightarrow\infty$ at finite ratio $N/M=r$ (<A HREF="http://en.wikipedia.org/wiki/Marchenko-Pastur_distribution">Marchenko-Pastur distribution</A>).