Vision-based motion capture solutions often struggle with occlusions, which result in the loss of critical joint information and hinder accurate 3D motion reconstruction. Other wearable alternatives also suffer from noisy or unstable data, often requiring extensive manual cleaning and correction to achieve reliable results. To address these challenges, we introduce the Masked Motion Diffusion Model (MMDM), a diffusion-based generative reconstruction framework that enhances incomplete or low-confidence motion data using partially available high-quality reconstructions within a Masked Autoencoder architecture. Central to our design is the Kinematic Attention Aggregation (KAA) mechanism, which enables efficient, deep, and iterative encoding of both joint-level and pose-level features, capturing structural and temporal motion patterns essential for task-specific reconstruction. We focus on learning context-adaptive motion priors, specialized structural and temporal features extracted by the same reusable architecture, where each learned prior emphasizes different aspects of motion dynamics and is specifically efficient for its corresponding task. This enables the architecture to adaptively specialize without altering its structure. Such versatility allows MMDM to efficiently learn motion priors tailored to scenarios such as motion refinement, completion, and in-betweening. Extensive evaluations on public benchmarks demonstrate that MMDM achieves strong performance across diverse masking strategies and task settings. The source code is available at https://github.com/jjkislele/MMDM.