To expand on my comments, this paper arxiv.org/pdf/hep-th/9209083v2.pdf by Shatashvili deals with ``correlation functions'' of Haar unitary matrices of the form $$ \int_{U(N)}^{} d\mu(U) e_{}^{tr(UAU_{}^{-1}B)} U_{i_1j_1}^{}\bar U_{k_1\ell_1}^{}\ldots U_{i_mj_m}^{}\bar U_{k_m\ell_m}, $$ and provides a certain combinatorial formula for these. Then setting $A=0$ would probably recover what you're asking about.
The same correlation functions (and an alternative formula for them) are also discussed in https://arxiv.org/pdf/hep-th/0502041.pdf