Computing $E[yy'Cyy']$ for normalized Gaussian $y$

Suppose $x\sim \text{Normal}(0,\Sigma)$ with diagonal $\Sigma$ . Is there a closed form solution or a good approximation to the following quantity where $y=x/\|x\|$ : $$f(C)=E[yy'Cyy']$$ Target is to understand the growth rate of $g(k)=\operatorname{Tr}(\underbrace{f\circ f \circ \cdots \circ f}_{k} \circ I)$ in terms of $k$ . For unnormalized $x$ , I can use Wick's theorem, and this iteration nicely factors, giving nice formulas for $g(k)$ . Wondering if there's a trick I can use to make it work for $y$ . Motivation: factoring density of $x$ is used in Bordelon paper to give loss curve of SGD. Extending this to $y$ will transfer this analysis to the Kaczmarz method.