Residual bound is an element of the perturbation analysis, which studies the effect of variation in data matrices on the solution and thus is obligatory for the interpretation of the computed in finite machine arithmetic environment solution. Residual bound is an effective practical stopping criterion when solving the equation by a numerically stable iterative algorithm. The aim of this paper is to derive residual bounds for the computed approximate solution to the nonlinear matrix equation \(X^p = A+M(B+X^{-1})^{-1})M^*\) , were \(p\ge 1\) is a positive integer, X, A, B and M are \(n \times n\) complex matrices. M is an arbitrary matrix, A and B are Hermitian positive semi-definite matrices, and the solution X is a Hermitian positive definite matrix. The bounds proposed in the paper are obtained, applying the techniques of the Fréchet derivatives, the method of Lyapunov majorants, and the Schauder fixed point principle and are limited to terms of second-order. A numerical example shows the efficiency of the bounds proposed in a wide range of values for the exponent p.