We theoretically and numerically investigate the effects of high-order dispersion (HOD) on microcavity solitons, both in time and frequency domain with an extended normalized Lugiato-Lefever equation (LLE). The observed temporal drift of bright and dark solitons is shown to originate from high-odd-order dispersion, while the sign determines the direction of soliton movement and the amplitude decides the drift speed. HOD can also be introduced to stabilize the breathing bright and dark cavity solitons. In spectral domain, the nonlinear symmetry breaking is mainly introduced by third-order dispersion, whereas both third- and fourth-order dispersion can introduce dispersive wave accompanied by soliton tail oscillation. This work could give insight for exploring detailed intracavity pulse dynamics and spectral characteristics of Kerr combs influenced by HOD, as well as provide a viable route to delicate control of Kerr comb generation through tailoring the dispersion parameters.