The answer is C: powerful.
The word ardent means a very strong or powerful feeling that is typically defined by an eager, intense, or fervent emotion, like in this case the children´s need at the sight of their father. Although there is little context in the excerpt to define the root of the children´s need, one can see that their reaction is strong enough to be characterized as ardent, passionate or powerful enough to make them almost fall out of the open door.
