The solution was obtained already by <A HREF="https://www.researchgate.net/publication/303008084_Distribution_of_eigenvalues_for_some_sets_of_random_matrices">Marchkenko and Pastur,</A> in terms of the Stieltjes transform $g(z)$ of the spectral density of $D+XX^{\rm T}$, see for example equation 3 of <A HREF="https://hal.archives-ouvertes.fr/hal-01346371/document">Spectrum of deformed random matrices and free probability:</A>

$$g(z)=\int\frac{1}{z-t[1-m/n+(mz/n)g(z)]}\rho(t)dt$$

where $\rho(t)$ is the spectral density of $D$. This holds for any random Hermitian perturbation $D$, irrespective of whether it is diagonal or not.