Let $X \sim \mathcal{W}_d(V, n)$ be a Wishart matrix and $A$ a real symmetric positive definite matrix.
I am interested in the following function $$ \phi(A) := \mathrm{Tr}\Big\{\mathbb{E}\big[\big(X + A \big)^{-1}\big] \Big\}. $$ I know that in general quantities like this should be related to the Stiltjes transformation associated to $X$, but I could not find the exact formula for this quantity.
The Stieltjes transform is given by the normalized trace of the resolvent $(X - I z)^{-1}$. With probability one, this is close, for every $z \in \mathbb{C}^+$, to the Stieltjes transform of the Marchenko-Pastur law associated with the spectral measure of $V$ when $n$ and $d$ are both large and comparable. A lot is known about the behavior of the Stieltjes transform as $z \rightarrow x \in \mathbb{R}$.
If $A$ is not a scalar multiple of the identity or constrained to have finite rank (rank is negligible with respect to d), I don't know any results for determining $\phi$. The only simplification that I can think of is that, if $V = I$, the distribution of $X$ is unitarily invariant, and so we can take $A$ to be diagonal without loss of generality. But, if $A$ isn't a multiple of the identity or has negligible rank w.r.t. d, I don't know how to proceed.