How does this exchange reflect the fact that he is impetuous?

Higgins is passive aggressive throughout the play. He inflicts pain and discomfort on Eliza, only to offer a semblance of "sympathy" later on. I think his offering of the handkerchief is to alleviate his own guilt rather than help Eliza. Higgins hence shows a condescending impetuousness to people that are weaker than him in social standing.