In the book "Spectral Analysis of Large Dimensional Random Matrices" by Bai and Silverstein, there is the following lemma:

Lemma B.26(pg. 530) Let $A=(a_{ij})$ be an $n\times n$ nonrandom matrix and $X = (x_1,\dots, x_n)$ be a random vector of independent entries. Assume that $\mathbb E x_i = 0, \mathbb E|x_i|^2 = 1,$ and $\mathbb E|x_j |^l\le \nu_l.$ Then, for any $p\ge 1$: $$\mathbb{E} | X^*AX - \text{tr} A |^p \le C_p \left( (\nu_4\text{tr}(AA^*))^{p/2} + \nu_{2p}\text{tr}(AA^*)^{p/2} \right).$$

Though not stated explicitly in the lemma, a look at the proof suggests that $C_p$ is given as at least exponential in $p$.

**Question**: Is there a more refined understanding of what $C_p$ is as a function of $p$? It seems to depend on several inequalities, including the Marcinkiewicz–Zygmund inequality, whose constants have been studied at least to some extent.