I have a partial answer.

Let $z \in \mathbb{R}^d$, let $A$ a positive definite $d\times d$ matrix, let $b > 0$, and let $q > (d-1)/2$. Define $p = q + (1-d)/2$ and $H = zz^{t}$. The following is claimed in the paper *"A Bayesian approach for clustering skewed data using
mixtures of multivariate normal-inverse Gaussian distributions"* by Yang & al.

Let $x \sim GIG(-p, z^tAz, b)$ (generalized inverse Gaussian) and $W \sim \mathcal{W}_d(q,A)$ (Wishart) be independant. Then ${(xH + W)}^{-1}$ has the $MGIG(-q, bH, A)$ distribution. 

So one has the expectation of *the inverse* of $MGIG(-q, bH, A)$. This is $\mathbb{E(x)}H + \mathbb{E}(W)$, and these two expectations are known.

Now I believe I saw somewhere that this inverse is also $MGIG$. I'll edit this answer if I retrieve the statement of this fact.

This is a very partial answer because $bH$ is of rank one and $-q < (1-d)/2 \leqslant 0$.