We introduce the notion of associative (BiHom-)Yang-Baxter pair of
weight
which can provide the solution to the
double curved Rota-Baxter (BiHom-)system.
Equivalent characterizations of (quasitriangular) covariant BiHom-bialgebra are given. We also prove that associative BiHom-Yang-Baxter equation of weight
can be obtained by
the unitary quasitriangular covariant BiHom-bialgebra. At last, we present two approaches to construct (BiHom-)pre-Lie modules from Rota-Baxter (BiHom-) paired modules.