Summary: Let \(R\) be a noncommutative prime ring with the extended centroid \(C\), \(I\) a nonzero ideal of \(R\) and \(g\) a \(b\)-generalized derivation of \(R\). We show that, if \([g(x^m),x^n]_k = 0\) for all \(x\in I\), where \(m\), \(n\), \(k\) are fixed positive integers, then there exists \(\lambda \in C\) such that \(g(x)=\lambda x\) for all \(x\in R\) unless \(R\cong M_2(\operatorname{GF}(2))\), the \(2\times 2\) matrix ring over the Galois field \(\operatorname{GF}(2)\) of two elements. This gives a natural generalization of the results for derivations, generalized derivations and generalized \(\sigma\)-derivations with an X-inner automorphism \(\sigma\).