Spatiotemporal fusion (STF) bridges the gap between temporal and spatial resolutions in satellite imagery, enabling effective monitoring of Earth's surface dynamics. However, existing methods rely on cloud-free reference images, a constraint that fails in realistic, cloud-prone scenarios. To overcome this, we propose the Cloud-Aware Wavelet Generative Adversarial Network (CLAW-GAN), a novel framework for high-fidelity reconstruction under cloud-contaminated conditions. CLAW-GAN introduces Regression Wavelet Analysis (RWA) to decouple spectral backgrounds from structural details. While a change-aware gated mechanism accounts for land-cover changes, the Frequency-Separated Fusion (FSF) module then independently integrates these components. To ensure visual realism, a multi-scale discriminator operates in the wavelet domain, enforcing consistency across high-frequency subbands to minimize artifacts. Evaluated on the newly introduced Global Cloud-shrouded Agricultural Regions (GCAR) benchmark and the simulated Daxing dataset, CLAW-GAN achieves state-of-the-art performance and demonstrates superior robustness across varying cloud coverage.