Is the wife's action of giving money she saved (which her husband gave her without his knowledge) to the needy, and which is later returned to her, considered a betrayal and a breach of trust, even though her intention is to relieve the distress of those in need?

It is not permissible to dispose of money given by a husband to his wife for safekeeping, because it is a trust (amanah) and a deposit (wadi'ah) with her. It is not permissible to dispose of it by lending or otherwise, except with his permission. Otherwise, it would be a betrayal of trust, based on the Almighty's saying: "O you who have believed, do not betray Allah and the Messenger or betray your trusts while you know [the consequence]," and based on the Prophet's (peace be upon him) saying: "There is no faith for one who has no trust, and no religion for one who has no covenant."