The double-cone ignition scheme is a promising novel ignition method, which is expected to greatly save the driver energy and enhance the robustness of the implosion process. In this paper, ablation of the inner surface of the cone by the hard X-ray from coronal Au plasma is studied via radiation hydrodynamics simulations. It is found that the X-ray ablation of the inner wall will form strong pre-plasma, which will significantly affect the implosion process and cause the Au plasma to mix with the fuel, leading to ignition failure. The radiation and pre-ablation intensities in the system are estimated, and the evolutions of areal density, ion temperature and the distribution of Au ions are analysed. In addition, the mixing of Au in CH at collision is quantified. Then, a scheme to reduce the X-ray pre-ablation by replacing the gold cone with a tungsten cone is proposed, showing that it is effective in reducing high-Z mixing and improving collision results.