Though the high-order superdirectivity theory proposed in recent years is attractive, it is hard to implement in practice due to its poor robustness to small random array errors. Hence, in this paper, we present two robust designs of high-order superdirectivity for circular arrays with gain and phase errors. Firstly, we study on the sensitivity function of the high-order superdirectivity and give an alternative solution for a robust superdirective beamformer based on sensitivity function constraint. This method could achieve an arbitrary compromise between directivity and robustness, so it is more flexible and applicable than the existing higher-order truncation method. Although it does not improve on computational complexity or performance with respect to the second-order cone programming obviously, it could lay the foundation for the following robust design method. Then, considering the fact that different eigenbeams correspond to different eigenvalues, we study the method of diagonal loading with variable factors in detail, and further improve the performance of the former sensitivity function constrained method by loading variable factors to different eigenbeams, which results in better performance and greater flexibility in making a compromise. We also show that this proposed loading variable factors method can achieve an equivalent result to the higher-order truncation method by setting proper factors. Simulation results demonstrate the robustness and effectiveness of the above two methods, especially the performance improvement of the loading variable factors method.