This is certainly not a complete answer, but only some suggestions on how to reduce the complexity of the problem.

It does not seem to me like the possible rationality of the integrand, depending on the value of $m$, would be of great help. Integrals of rational functions are likely to produce non-rational ones (logarithms, roots, etc.). So computing the coordinate integrals one by one seems hopeless. Also, using global methods, like residues also seems problematic because the integration domain is restricted to the first orthant.

On the other hand using the SVD decomposition of $M$ puts the integrand into canonical form, where the square of the norm becomes a sum of squares, without cross terms between the integration coordinates. The downside is that the integration domain is no longer the first orthant, but rather a rotated and stretched version thereof.

The level sets of this new integrand are then concentric spheres, with center determined by the vector $C$. This symmetry can be exploited by changing to spherical coordinates with the same center. The Integrand then becomes $\int_0^\infty \mathrm{d}r~ r^{n-1} (r^2+c^2)^{-m/2} \int_{P_r} \mathrm{d}\Omega$, where $c^2$ is the least squares residual determined by $C$, $\mathrm{d}\Omega$ is the angular integration $P_r$ is the curved $n$-simplex obtained by intersecting the $n$ planes delimiting your integration domain with the sphere of radius $r$. The entire complexity of the integration reduces to computing the volume of $P_r$ as a function of $r$.

Unfortunately, this volume computation is in itself likely to be rather non-trivial. I do know that the computation of volumes of spherical simplices is a classical subject, treated early on by Schläfli. However, not being very familiar with the subject, I wouldn't know how to find the exact formulas. This survey article by Vinberg may be of help, though. An extra complication is that your $P_r$ are not precisely spherical simplices, because their sides do not correspond to higher dimensional "great circles", unless one of the planes delimiting your integration domain passes through the origin. However, the $P_r$ do approach true spherical simplices in the limit of large $r$. If you were able to find a decent analytic approximation for the volume of $P_r$ for small $r$, you may be able to get approximate analytic control over the whole integral. But I don't know how to do that.