This paper introduces a novel framework for image and video demoiréing by integrating Maximum A Posteriori (MAP) estimation with advanced deep learning techniques. Demoiréing addresses inherently nonlinear degradation processes, which pose significant challenges for existing methods. Traditional supervised learning approaches either fail to remove moiré patterns completely or produce overly smooth results. This stems from constrained model capacity and scarce training data, which inadequately represent the clean image distribution and hinder accurate reconstruction of ground-truth images. While generative models excel in image restoration for linear degradations, they struggle with nonlinear cases such as demoiréing and often introduce artifacts. To address these limitations, we propose a hybrid MAP-based framework that integrates two complementary components. The first is a supervised learning model enhanced with efficient linear attention Test-Time Training (TTT) modules, which directly learn nonlinear mappings for RAW-to-sRGB demoiréing. The second is a Truncated Flow Matching Prior (TFMP) that further refines the outputs by aligning them with the clean image distribution, effectively restoring high-frequency details and suppressing artifacts. These two components combine the computational efficiency of linear attention with the refinement abilities of generative models, resulting in improved restoration performance.