Is it permissible to supplicate for marriage to a married man with a sincere intention of not committing a sin or severing ties of kinship?

There is no harm for a woman in supplicating to Allah to marry her to a specific married man. The permissibility of seeking marriage with a married man is evidenced by Umar's offer of his daughter to Abu Bakr, and then the Prophet, peace and blessings be upon him, married her, while both of them were married. Also, a woman offered herself to the Prophet, peace and blessings be upon him, and he was married at the time.