Is the hijab that does not cover the face considered incomplete, and does the one who wears it receive reward?

A Muslim woman must wear the full hijab (definition), including covering her face. This is one of Allah Almighty’s blessings that preserves her chastity and modesty, and protects her from harm and the covetousness of evildoers, because the face is the center of temptation. Allah Almighty says: (O Prophet! Tell your wives and your daughters and the women of the believers to draw their cloaks all over their bodies (i.e. screen themselves completely except the eyes or one eye to see the way). That will be better, that they should be known as righteous women, so as not to be annoyed. And Allah is Ever Oft-Forgiving, Most Merciful).