Spectrally-weighted Stieltjes transform of random matrix $Z=XX^\top$ in terms of Stieltjes transform of $Z$ and the weighting function

Question

Let $n$ and $d$ positive integers going to infinity such that $d/n \to \gamma \in (0,\infty)$. Let $X$ be a random $n \times d$ iid rows from $N(0,\Sigma)$, where $\Sigma = diag(\lambda_1,\ldots,\lambda_d)$ with $1/M \le \lambda_j \le M$, for all $j \in [d]$ and for some fixed $M \in (0,\infty)$.

Let $w:(0,\infty) \to (0,\infty)$ be a fixed function and let $C := w(\Sigma) = diag(w(\lambda_1),\ldots,w(\lambda_n))$. For any fixed $\lambda>0$, define $T(\lambda)$ by $$ T(\lambda) := (1/d)\mbox{trace}((G + \lambda I_n)^{-1} G (G+\lambda I_n)^{-1} H), $$ where $G=XX^\top/d$ and $H := XCX^\top/d$, random $n \times n$ psd matrices.

Question. Is there an expression for the limiting value of $T(\lambda)$ in terms of the Stieltjes transform (say) of $G$ ?