Is a marriage contract valid when its two Muslim witnesses do not pray, and one of them occasionally drinks alcohol, and if it is invalid, what should the spouses do after having had two children?

The Fatwa's Summary:

According to the majority of scholars, marriage is invalid with the testimony of a transgressor. However, the Hanafis do not stipulate uprightness ( 'adalah) for marriage witnesses. Therefore, the marriage contract, with the testimony of the two men mentioned, knowing their condition, is invalid according to the majority. Nevertheless, children born from such a union are attributed to their father, because lineage is established in any marriage that the couple believed to be valid.