Deploying Large Language Models (LLMs) in real-world dynamic environments raises the challenge of updating their pre-trained knowledge. While existing knowledge editing methods can reliably patch isolated facts, they frequently suffer from a "Reasoning Gap", where the model recalls the edited fact but fails to utilize it in multi-step reasoning chains. To bridge this gap, we introduce MCircKE (\underline{M}echanistic \underline{Circ}uit-based \underline{K}nowledge \underline{E}diting), a novel framework that enables a precise "map-and-adapt" editing procedure. MCircKE first identifies the causal circuits responsible for a specific reasoning task, capturing both the storage of the fact and the routing of its logical consequences. It then surgically update parameters exclusively within this mapped circuit. Extensive experiments on the MQuAKE-3K benchmark demonstrate the effectiveness of the proposed method for multi-hop reasoning in knowledge editing.