The influence of the built-in electric field on bound polaron binding energy and the polaron shift in a wurtzite ZnO/MgxZn1-xO quantum well are investigated using the improved Lee-Low-Pines (LLP) variational theory. The ground-state energy and binding energy, the contributions from different branches of optical phonons to the energy and the binding energy are given as the functions of composition x. The anisotropic properties of dielectric constant, electron band mass, different branches of optical phonons frequencies, the optical phonon-electron and the impurity center interaction are considered in the numerical calculations. The results show that the influence of the built-in electric field on the binding energy and polaron shift is obvious, and the degree of the influence of the built-in electric field for the contributions of different phonon modes is different. The built-in electric field increases the total phonon contribution to the energy, but it significantly reduces the total phonon contribution to the binding energy. The binding energy with the built-in electric field rapidly decreases as increasing composition x, but the binding energy without the built-in electric field decreases slightly. The result also show that when increasing the composition x, the contributions of interface and confined phonons to the energy and the binding energy with and without the built-in electric field increase, the contributions of half space phonon reduce, and the total contributions of phonons to the energy increase. But the total contributions of phonons to the binding energy with and without the built-in electric field are different. The total contributions with the built-in electric field increase, while the total contributions without the built-in electric field decrease. In comparison with the zinc blende GaAs/AlxGa1-xAs quantum wells, the influence of optical phonons on the energy and the binding energy of bound polaron in a wurtzite ZnO/MgxZn1-xO quantum well is larger, and polaron shift is more obvious.