Semenov-Tian-Shansky has introduced the modified classical Yang-Baxter equation, which is called the modified $r$-matrix. Relevant studies have been extensive in recent times. This paper is devoted to study cohomology theory of modified Rota-Baxter pre-Lie algebras and its applications. First we introduce the concept and representations of modified Rota-Baxter pre-Lie algebras. We then develop the cohomology of modified Rota-Baxter pre-Lie algebras with coefficients in a suitable representation. As applications, we consider the infinitesimal deformations, abelian extensions and skeletal modified Rota-Baxter pre-Lie 2-algebra in terms of lower degree cohomology groups.

Uchino first initiated the study of generalized Reynolds operators on associative algebras. Recently, related research has become a hot topic. In this paper, we first introduce the notion of generalized Reynolds operators on Hom-Lie triple systems associated to a representation and a 3-cocycle. Then, we develop cohomology of generalized Reynolds operators on Hom-Lie triple systems with coefficients in a suitable representation. As applications, we use the first cohomology group to classify linear deformations and we study the obstruction class of an extendable order $n$ deformation. Finally, we introduce and investigate Hom-NS-Lie triple system as the underlying structure of generalized Reynolds operators on Hom-Lie triple systems.