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In this paper, we focus on the Moore–Penrose metric generalized inverse of the modified operator B=A+UGV, where A,U,G,V are bounded linear operators between some Banach spaces. We establish conditions that guarantee the existence of the Moore–Penrose metric generalized inverse of B. As a consequence, we present an extension of the so-called classical Sherman–Morrison–Woodbury formula for the Moore–Penrose metric generalized inverse. Some particular cases and applications will be also considered. [ABSTRACT FROM AUTHOR]