Capturing display screens with mobile devices has become increasingly common, yet the resulting images often suffer from severe degradations caused by the coexistence of moiré patterns and flicker-banding, leading to significant visual quality degradation. Due to the strong coupling of these two artifacts in real imaging processes, existing methods designed for single degradations fail to generalize to such compound scenarios. In this paper, we present the first systematic study on joint removal of moiré patterns and flicker-banding in screen-captured images, and propose a unified restoration framework, named CLEAR. To support this task, we construct a large-scale dataset containing both moiré patterns and flicker-banding, and introduce an ISP-based flicker simulation pipeline to stabilize model training and expand the degradation distribution. Furthermore, we design a frequency-domain decomposition and re-composition module together with a trajectory alignment loss to enhance the modeling of compound artifacts. Extensive experiments demonstrate that the proposed method consistently. outperforms existing image restoration approaches across multiple evaluation metrics, validating its effectiveness in complex real-world scenarios.