Given a query, how can a QA system ask an informative clarifying question to accurately answer it? Open-domain conversational question answering requires not only detecting when a query is ambiguous but also asking clarifying questions that efficiently uncover the latent intent of the user. However, existing approaches either rely on prompt-only clarify-or-answer decisions with limited robustness or distill clarification trajectories without explicitly modeling intent convergence. In this paper, we propose D4CQ (Data-Efficient Diagnosticity based Dynamic Distillation for Clarifying Question), a data-efficient framework for learning diagnostic clarifying questions through dynamic distillation. Our approach distills a structured refinement protocol, prioritizes informative clarifications via diagnosticity-based reweighting, and improves sample efficiency through hindsight relabeling. Empirical results demonstrates that D4CQ consistently outperforms strong baselines, while maintaining comparable interaction budgets. Our code is available at https://github.com/ddddcq/d4cq.