Accurately retrieving relevant bid keywords for user queries is critical in Sponsored Search but remains challenging, particularly for short, ambiguous queries. Existing dense and generative retrieval models often fail to capture the nuanced user intent in these cases. To address this, we propose an approach to enhance query understanding by augmenting queries with rich contextual signals derived from web search results and large language models, stored in an online cache. Specifically, we use web search titles and snippets to ground queries in real-world information, and utilize GPT-4 to generate query rewrites and explanations that clarify user intent. These signals are efficiently integrated through a Fusion-in-Decoder based Unity architecture, enabling both dense and generative retrieval with serving costs on par with traditional context-free models. To address scenarios where context is unavailable in the cache, we introduce context glancing, a curriculum learning strategy that improves model robustness and performance even without contextual signals during inference. Extensive offline experiments demonstrate that our context-aware approach substantially outperforms context-free models. Furthermore, online A/B testing on a prominent search engine across 160+ countries shows significant improvements in user engagement and revenue.