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Open-domain question answering (OpenQA) is an important branch of textual QA which discovers answers for the given questions based on a large number of unstructured documents. Effectively mining correct answers from the open-domain sources still has a fair way to go. Existing OpenQA systems might suffer from the issues of question complexity and ambiguity, as well as insufficient background knowledge. Recently, conversational OpenQA is proposed to address these issues with the abundant contextual information in the conversation. Promising as it might be, there exist several fundamental limitations including the inaccurate question understanding, the coarse ranking for passage selection, and the inconsistent usage of golden passage in the training and inference phases. To alleviate these limitations, in this paper, we propose a framework with Multifaceted Improvements for Conversational open-domain Question Answering (MICQA). Specifically, MICQA has three significant advantages. First, the proposed KL-divergence based regularization is able to lead to a better question understanding for retrieval and answer reading. Second, the added post-ranker module can push more relevant passages to the top placements and be selected for reader with a two-aspect constrains. Third, the well designed curriculum learning strategy effectively narrows the gap between the golden passage settings of training and inference, and encourages the reader to find true answer without the golden passage assistance. Extensive experiments conducted on the publicly available dataset OR-QuAC demonstrate the superiority of MICQA over the state-of-the-art model in conversational OpenQA task.