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We compute the deterministic approximation for mixed fluctuation moments ofproducts of deterministic matrices and general Sobolev functions of Wignermatrices. Restricting to polynomials, our formulas reproduce recent results of[Male, Mingo, Pech\'e, Speicher 2022], showing that the underlyingcombinatorics of non-crossing partitions and annular non-crossing permutationscontinue to stay valid beyond the setting of second-order free probabilitytheory. The formulas obtained further characterize the variance in thefunctional central limit theorem obtained recently in the companion paper[Reker 2023].

Fluctuation Moments for Regular Functions of Wigner Matrices