Large language models (LLMs) show strong performance in medical question answering, but their computational costs limit deployment under resource constraints. Therefore, an increasing number of studies focus on distilling knowledge from LLMs into smaller models to improve performance in medical question answering. However, existing methods often fail to identify truly useful knowledge to distill, making it difficult to focus distillation on the student’s real weaknesses. As a result, much of the distillation budget is spent on already acquired knowledge rather than on weaknesses that still limit performance. To address this problem, we propose LEAD, a Learnable Error-Aware Distillation framework that precisely improves the weaknesses of student models. First, LEAD identifies questions answered incorrectly by the student model to determine which weaknesses require teacher guidance. Then, LEAD generates teacher rationales and similar samples to construct candidate supervision for these errors. Through supervision refinement, reliable and learnable training samples are retained from the candidate supervision. Finally, LEAD trains the student on a balanced sample mixture to correct learnable errors while preserving previously acquired knowledge. Experiments on four medical question answering benchmarks show that LEAD outperforms existing distillation methods.