Is an expiatory offering (kaffarat yamin) obligatory upon someone who swore not to help his daughter and her husband, but then helped them financially after the husband's religious condition improved and he was expelled from his family's home?

We praise Allah for guiding your daughter's husband. Registering the 'urfi marriage contract in court is a benefit for preserving and documenting rights. As for breaking your vow not to assist him in this marriage, scholars have differed on the wording "I covenant with Allah concerning such and such." The preponderant opinion for matters other than acts of obedience is that it is an oath requiring an expiation for an oath (kaffarat yamin). The Hanafis and Hanbalis held that it is an oath, while the Malikis have two opinions, and for the Shafi'is, it is not an oath unless intended as such. Sheikh al-Islam Ibn Taymiyyah elaborated that it can be both a vow (nadhr) and an oath if one commits to it as an act of nearness to Allah and obedience, and it is only an oath if one commits to something that is not an act of nearness. Therefore, you are obligated to offer an expiation for an oath, and you should increase your acts of obedience.