Is the mother still legally married to her husband, given the father's absence and refusal to divorce, and her desire to marry another man?

If the father is characterized by what has been mentioned, the mother should have sought divorce or khul’ from him previously. Since she did not do so, she is still his wife, and it is not permissible for her to marry another without a divorce. However, she has the right to approach the Sharia judge to take up her case and rule in her favor for a divorce. After obtaining it and the expiry of her iddah, she is permitted to marry. Until then, she remains a wife, and she must be patient and seek help from Allah.