Large Language Models (LLMs) have made significant strides but struggle in knowledge-intensive fields like Medical Visual Question Answering (Med-VQA). Medical reasoning with LLMs often suffers from hallucination due to the specialized nature of medical imagery and queries. In this paper, we propose the Data and Knowledge Adaptive Medical Reasoning (DKA-MR) framework to bridge LLMs and Knowledge Graphs (KGs) for a comprehensive understanding of the visual clues behind medical imagery and queries. At the core of DKA-MR is the Graph-to-Chain Medical Reasoning (GCMR) module, which utilizes a Prompt-driven Augmentation strategy to extract the core entities from the visual explanations of medical images and detects highly relevant knowledge-chain prompts from medical KGs for LLMs to reduce hallucinations. By aligning knowledge-chains with input queries, this approach not only enhances answer interpretability and traceability but also enabling human-like, step-by-step reasoning grounded in structured medical knowledge. Additionally, we introduce the Med-Knowledge Enhanced CutMix module, which fuses visual and textual features to create mixed images, enriching the training data and capturing intricate semantic relationships between medical images and their descriptions. Extensive experiments on the VQA-RAD and SLAKE datasets demonstrate that DKA-MR achieves state-of-the-art performance in handling both closed-ended and open-ended questions.