  A LieYRep pair consists of a Lie-Yamaguti algebra and its representation. In
this paper, we establish the cohomology theory of LieYRep pairs and
characterize their linear deformations by the second cohomology group. Then we
introduce the notion of relative Rota-Baxter-Nijenhuis structures on LieYRep
pairs, investigate their properties, and prove that a relative
Rota-Baxter-Nijenhuis structure gives rise to a pair of compatible relative
Rota-Baxter operators under a certain condition. Finally, we show the
equivalence between $r$-matrix-Nijenhuis structures and Rota-Baxter-Nijenhuis
structures on Lie-Yamaguti algebras.
