In drug discovery, accurately predicting molecular activity is crucial for identifying and optimizing molecules with desirable biological properties. A significant challenge in this field is the phenomenon of activity cliffs, where molecules with similar structures exhibit significantly divergent biological activities. This study introduces MAPCliff-WMGR, a computational framework designed to predict molecular activity under the activity cliff scenario using weighted molecular graphs. MAPCliff-WMGR consists of a core mGraphSNNGAT module that integrates model-specific adjustments to better handle molecular data, enabling the model to effectively predict molecular activity under the activity cliff scenario. Due to activity cliff data exhibiting characteristics of spectral bias, MAPCliff-WMGR addresses this by employing an Independent Feature Mapping (IFM) module that uses sinusoidal transformations to map features into a frequency-rich domain. Experimental results demonstrate that MAPCliff-WMGR achieves an average RMSE of 0.677 for cliff molecules, which is 7.2% better than the best-performing baseline. Furthermore, we build the MACE-R7 platform, a richer benchmark with various response types and targets, on which our method achieves an average improvement of 3.2% in overall prediction and 8.7% for cliff molecules. Moreover, the model's interpretability is further demonstrated to uncover critical atoms responsible for activity cliffs through attention-based analysis and dimensionality reduction visualizations. Finally, a case study on small-molecule drugs targeting estrogen receptor alpha (ERα) for breast cancer treatment underscores the model's ability to accurately predict activity for cliff molecules, validating its potential for virtual drug screening.