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 childrenĀ“s need at the sight of their father. Although there is little context in the excerpt to define the root of the childrenĀ“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.