In this paper, we present the results of our research on the boundary value problem model for fourth-order nonlinear differential equations of the Kirchhoff type. \(z^{\left( 4 \right)} \left( t \right) = K\left( {\mathop \smallint \nolimits_{0}^{1} \left( {z^{\prime}\left( s \right)} \right)^{2} ds} \right)z^{\prime\prime}\left( t \right) + f\left( {t, z\left( t \right), z^{\prime}\left( t \right), z^{\prime\prime}\left( t \right), z^{\prime\prime\prime}\left( t \right)} \right), 0 < t < 1,\) with mixed boundary condition. Using the Green's function method, we transform the problem into an operator equation for the right-hand side. From there, we investigate the conditions for the existence and uniqueness of solutions for the right-hand side and propose a continuous iterative method to find the solution. By utilizing high-accuracy differential and definite integration methods, we construct a discrete iterative method to approximate the solution of the problem with eighth-order accuracy. Computational results obtained from specific examples confirm the theoretical correctness as well as the accuracy of the proposed method.