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This work encompasses the study of certain weak maps termed as weak right η-centralizers defined on [Formula: see text] a noncommutative prime ring to [Formula: see text]. The first section provides an overview of the motivation behind our work. In the second section, our main focus is to establish the characterization of weak right η-centralizers while addressing the theory of functional identities, particularly in the challenging case of low dimensions. Furthermore, in the third section, we derive the characterization of weak Jordan σ-derivations based on the aforementioned characterization.