This paper proposes Dynamically Adaptive MCTS-based Reasoning (DAMR), a novel framework for Knowledge Graph Question Answering (KGQA). DAMR integrates LLM-based Monte Carlo Tree Search (MCTS) with adaptive path evaluation to enable efficient and context-aware KGQA. Based on MCTS, DAMR effectively reduces the search space by selecting the top k semantically relevant relations at each expansion step through an LLM-based planner. Furthermore, it introduces a lightweight Transformer-based score that jointly encodes question and relation sequences via cross-attention, thereby capturing subtle semantic changes during multi-hop inference and improving evaluation accuracy. Furthermore, to mitigate the lack of high-quality supervision, DAMR incorporates a dynamic pseudo-path refinement mechanism that periodically generates training signals from partial paths explored during search, allowing the score to continuously adapt to the evolving inference trajectory distribution. Extensive experiments on several KGQA benchmarks demonstrate that DAMR significantly outperforms state-of-the-art methods.