How can one genuinely repent of the sin of marital infidelity and seek Allah's forgiveness after the wife's supplication for hardship in provision was answered, given the impossibility of confronting the wife for fear of the family's collapse?

The door of repentance is open, and its conditions are: abandoning the sin, regretting it, and resolving not to return to it. This should be done for the sake of Allah, not out of fear of one's wife. You are not required to inform your wife of what you did; rather, you should conceal yourself, as the Prophet (peace and blessings of Allah be upon him) said: "Whoever is afflicted with any of these impurities should conceal himself with the concealment of Allah, the Exalted and Glorious." It is recommended to associate with righteous people and avoid means of disobedience, as this will improve your condition and facilitate matters. You should praise Allah for alerting you to repentance before death.