The solution was obtained already by Marchkenko and Pastur, in terms of the Stieltjes transform $g(z)$ of the spectral density of $D+XX^{\rm T}$, see for example equations 2 + 3 of Spectrum of deformed random matrices and free probability:
$$g(z)=\int\frac{1}{z-t-(m/n)(z-t)g(z)-1+m/n]}\rho(t)dt$$
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.