This paper proposes a movable antenna (MA) array-aided beam coverage scheme for low-earth orbit (LEO) satellite communications to enhance beamforming and interference mitigation. The key idea is to jointly optimize the antenna position vector (APV) and antenna weight vector (AWV) of the satellite-mounted MA array over time to minimize signal leakage to the interference area while ensuring sufficient beamforming gain over the coverage area. The satellite's orbit and coverage requirements are considered, and the APV and AWV are optimized to adapt to the time-varying coverage needs of terrestrial users. The continuous-time decision process is transformed into a discrete-time optimization problem, and an alternating optimization (AO)-based algorithm is developed using successive convex approximation (SCA) to obtain locally optimal solutions. Additionally, a low-complexity MA scheme is proposed by using an optimized common APV over all time slots, reducing antenna movement overhead. Simulation results show that the proposed MA array-aided beam coverage schemes significantly reduce interference leakage compared to conventional fixed-position antenna (FPA)-based schemes, while the low-complexity MA scheme achieves performance comparable to the continuous-movement scheme. The main contributions include the formulation of a non-convex optimization problem for MA array-aided beam coverage, the transformation of the problem into a tractable discrete-time optimization problem, and the development of an AO-based algorithm with SCA for optimization. The low-complexity MA scheme is also proposed to reduce computational complexity and movement overhead. The proposed scheme is validated through simulations, demonstrating its effectiveness in enhancing beam coverage and reducing interference leakage in LEO satellite communications.This paper proposes a movable antenna (MA) array-aided beam coverage scheme for low-earth orbit (LEO) satellite communications to enhance beamforming and interference mitigation. The key idea is to jointly optimize the antenna position vector (APV) and antenna weight vector (AWV) of the satellite-mounted MA array over time to minimize signal leakage to the interference area while ensuring sufficient beamforming gain over the coverage area. The satellite's orbit and coverage requirements are considered, and the APV and AWV are optimized to adapt to the time-varying coverage needs of terrestrial users. The continuous-time decision process is transformed into a discrete-time optimization problem, and an alternating optimization (AO)-based algorithm is developed using successive convex approximation (SCA) to obtain locally optimal solutions. Additionally, a low-complexity MA scheme is proposed by using an optimized common APV over all time slots, reducing antenna movement overhead. Simulation results show that the proposed MA array-aided beam coverage schemes significantly reduce interference leakage compared to conventional fixed-position antenna (FPA)-based schemes, while the low-complexity MA scheme achieves performance comparable to the continuous-movement scheme. The main contributions include the formulation of a non-convex optimization problem for MA array-aided beam coverage, the transformation of the problem into a tractable discrete-time optimization problem, and the development of an AO-based algorithm with SCA for optimization. The low-complexity MA scheme is also proposed to reduce computational complexity and movement overhead. The proposed scheme is validated through simulations, demonstrating its effectiveness in enhancing beam coverage and reducing interference leakage in LEO satellite communications.