This paper investigates Nikolskii and Besov function spaces defined using \(L_p\) -averaged mixed moduli of continuity of appropriate orders, instead of relying on classical moduli of continuity tied to specific mixed derivatives. We provide upper and lower bounds for the optimal accuracy of derivative reconstruction based on function values at a fixed number of points, for functions belonging to these classes on domains of a certain type. These bounds are either comparable or tighter than previously established results by the author for such function classes on the unit cube \(I^d\) . This work also significantly extends the range of Nikol’skii and Besov spaces with mixed smoothness for which such estimates have been derived.