Is what the questioner does—unintentionally looking at a girl's face that suddenly appears in video clips, then averting the gaze—considered extending the gaze, even with his eagerness to follow what is beneficial and avoid prohibitions?

The statement that a woman's face is not awrah (private part that must be covered) and therefore permissible to look at is not a matter of consensus but rather one of dispute. A group of jurists has held that it is awrah and impermissible to look at, and this is the fatwa (definition) (religious ruling) adopted by us. Therefore, you should avert your gaze from looking at the faces of women in programs, on roads, in public places, and elsewhere.