By using the homotopy analysis method (HAM), the electrostatic potential distribution problems of a type of high-order weakly nonlinear composite with a cylindrical inclusion randomly embedded in a host medium, which obeyes a current-field constitutive relation of J = σ E + χ |E|2E + η|E|4E, are investigated under the action of an external direct current electric field. With the mode expansion method, the current-field constitutive relation and their boundary conditions are transformed into a series of boundary value problems of nonlinear ordinary differential equations. Then the HAM is used to solve the boundary value problems of nonlinear ordinary differential equations and the asymptotic analytical solutions of electrostatic potential distribution in the inclusion and the host regions are given.