What is the ruling on maintaining ties with a father who abandoned his children for 20 years and did not spend on them, knowing that the mother is harmed by these ties?

Your mother does not have the right to prevent you from honoring your father and showing kindness to him, as this is your duty. The father's right is not forfeited by his injustice or mistreatment. Do not obey your mother if she forbids your father from visiting you. Continue to honor him, and strive to appease your mother and deal with her kindly by not informing her of your father's visits if possible. Otherwise, apologize to her by explaining that you are commanded by Islamic law to honor your father.