Am I obligated to give my father a fixed monthly sum, knowing that he is well-off and likes me to give him, despite the fact that I was in hardship and no one helped me, and my wife objects, arguing that he does not need the money and that he did not help us in the past?

If your father is not in need, then you are not obligated to give him a portion of your wealth. However, honoring one's parents is a great matter in the sight of Allah, and Allah has associated gratitude to Him with gratitude to them and treating them with kindness. The Prophet, peace and blessings be upon him, said: "You and your wealth belong to your father," and he encouraged honoring one's parents to attain Paradise. Therefore, strive to please your father, and if that can only be achieved through money, then hasten to provide it. Be wary of your wife, who might dissuade you from this, for Allah says: (O YOU WHO HAVE BELIEVED, INDEED, AMONG YOUR WIVES AND YOUR CHILDREN ARE ENEMIES TO YOU, SO BEWARE OF THEM). Remember that your father raised you and spent on you, whereas your wife knew you after you had reached your maturity.