Let $A$ be an $n \times n$ random symmetric matrix with $E(A_{i j}) = 0$, $Var(A_{i j}) = 1/n$ for any $i,j$. $B$ is an $n \times n$ symmetric matrix with $B_{ii} = 0$.

I need to find a upper bound of $Tr(A^2 (A\circ B))$ as tight as possible. Notation $\circ$ is Hadamard product and $Tr(\cdot)$ is the trace of a matrix.

Currentely, my approach is as follows. According to von Neumann's trace inequality,

$Tr(A^2 (A\circ B)) \le \sum_i \sigma_i(A^2) \sigma_i (A \circ B) = \sum_i \big(\sigma_i (A)\big)^2 \sigma_i (A \circ B)$ ,

where $\sigma_i(\cdot)$ is the $i$th singular value of a matrix.

However, I do not know how to handle with $\sigma_i (A \circ B)$. Is there any way to deal with this or is there any better idea to find the upper bound?