Let $a \sim \mathcal{N}(0,I_m)$ be independent and $C \in \mathbb{R}^{m \times n}$ have iid $\mathcal{N}(0,1)$ entries. Fix $x \in \mathbb{R}^n$. I am interested in calculating $$\mathbb{E}\left[|\langle a,Cx \rangle|^2\right].$$
By rotational invariance, I believe we may take $x = e_1$. Then the problem reduces to calculating $$\mathbb{E}\left[\langle a, c \rangle^2\right]$$ where $c$ is the first column of $C$ and follows $\mathcal{N}(0,I_m).$ I am not sure how to compute the squared inner product expectation. Without the square, we have by independence $$\mathbb{E}\left[\langle a,c \rangle\right] = \sum_{i=1}^m \mathbb{E}[a_i \cdot c_{i}] = 0.$$ Any help on the squared calculation would be greatly appreciated!
You can write your expectation as a nested integral. On the outside, expectation with respect to $c$. On the inside, expectation with respect to $a$. You can re-express the number $|\langle a,c\rangle|^2$ as the matrix product $c' a a' c$, for which the expectation over $a$ works out to $c' c$. You can (I hope) finish the job without trouble.