Disclaimer. I only started learning the subject of free probability $1$ day ago, and I'm still trying to absorb the fundamentals, while applying them to my own specific problems arizing in the spectral analysis of certain concrete random matrices.

Let $X_{n,m}$, $Y_{m,k}$, $Z_{m,k}$ be large independent random matrices with entries from $N(0,1)$ and let $A_{m,k}$ be a deterministic matrix. Consider the random psd matrix $R_{m,k} := (X_{n,m}Y_{m,k}+Z_{m,k}+A_{m,k})(X_{n,m}Y_{m,k}+Z_{m,k}+A_{m,k})^\top$.

Question. How to use tools from free probability (e.g the "linearization trick", etc.) to compute the limiting spectral distribution of $R_{m,k}$ ?