My mother is asking me to do something that makes things hard for my wife. Whom should I put first?

Devotion to your mother is obligatory, but it does not permit injustice towards your wife; find a way that fulfils both rights. The Prophet ﷺ said: “Obedience is only in what is right (ma'ruf)” (Sahih al-Bukhari 7257), so a request that wrongs someone need not be obeyed. If your mother's request is reasonable, carry it out well. If it harms your wife, decline gently and look for a middle way without being disrespectful. Explain to your mother respectfully, and reassure your wife that her rights are protected. Wisdom, not blind taking of sides, is what resolves situations like this. See in-laws.