Let $G = U(n)$ and let $\rho, \pi$ be two irreducible unitary representations of $G$ on vector spaces $V$ and $W$, respectively. We know that
$$\int \rho(g)_{ij} \pi(g^{-1})_{kl} dg = \delta_{\rho \pi} \delta_{ik}\delta_{jl}/d_{\rho},$$
where $d_{\rho} = \dim(V)$ and $\delta_{\rho \pi}$ means that the integration gives zero if $\pi \neq \rho$.
Are there analogous formulas for the integration of higher powers of the matrix elements $\rho(g)_{ij}$ and $\pi(g)_{kl}$, i.e., can one compute integrals like
$$\int \rho(g)_{ij}\rho(g^{-1})_{i'j'}\pi(g)_{kl}\pi(g^{-1})_{k'l'}dg,$$
and so on?