Unmanned vessels, essential for ocean observation, often utilize an active hydrofoil propulsion system. When a vessel moves in waves, the tilted swing arm causes the motion trajectory of the hydrofoil to be asymmetric. This study employs a turbulence model with a Reynolds number of 105, simulated using the finite volume method (FVM) and the Navier-Stokes equations. Numerical simulations are then used to analyze the effects of the swing arm amplitude, arm length, and asymmetry ratio on the propulsive performance of a full-active hydrofoil with an asymmetric arc-trajectory. This asymmetrical arc-trajectory has a significant impact on the propulsion performance of the hydrofoil, especially when the asymmetry ratio is large. When the asymmetry ratio is 0.66, the mean thrust coefficient of the hydrofoil decreases by 68.37%, compared to the symmetric case. We further point out that such decrease is achieved through changes in foil flapping velocities and foil-vortex interactions between unequal down-stroke and up-stroke phases. However, when the asymmetry ratio is small, the impact of the asymmetric arc-trajectory can be almost negligible. Additionally, we find that the propulsion performance of the hydrofoil increases with increasing swing arm amplitude and arm length. These findings not only provide insights toward the hydrodynamics characteristics of the hydrofoil in asymmetric arc-trajectory mode, but also offer a reference to the design of the high-performance flapping hydrofoil propulsion system.