What are the available options for marrying a man who is already married with children after the mother's refusal, and is it necessary to speak with the father, and if the father refuses, is it permissible to marry with the Imam as a guardian?

If the suitor is pleasing in his religion and character, and you wish to marry him, ask him to propose to your guardian (your father), and mention his good qualities to your father. If your father agrees, then الحمد لله (praise be to Allah). If he refuses for an acceptable reason, then it is not permissible for you to seek marriage with him through an Imam or anyone else, because the guardian is a condition for the validity of the marriage, as stated in the hadiths: "There is no marriage except with a guardian," and "Any woman who marries without the permission of her guardian, her marriage is invalid." It is not permissible to bypass the guardian unless he is preventing the marriage (عاضلاً), and 'adl (prevention) is when a woman is prevented from marrying a suitable man whom she has accepted.

If 'adl is proven, guardianship transfers to the next in line after the father, such as a brother or grandfather. If none of them exist or they refuse, then the Muslim judge or the Imam of the mosque may marry her.

However, a guardian's refusal to marry off a girl to a married man with children is not considered 'adl, especially if the guardian believes he is not suitable for her, or fears harm or problems for her with the first wife. The guardian considers the best interest of his ward. Moreover, the Shari'ah has granted him this right, an example of which is the Prophet's (peace be upon him) refusal to marry Fatimah to Abu Bakr and Umar due to their old age.

So, do not object to your parents if they refuse this suitor, for they are more discerning in marital life and more eager for your benefit.