There were times, listening to Arnold and narrowing his eyes in the same heat-struck gaze, when Jack felt turned about.
What is the function of present participle here?
And why is the restrictive relative clause separated from its antecedent?
There were times, [listening to Arnold and narrowing his eyes in the same heat-struck gaze], when Jack felt turned about.
There are two ing-form participles here, "listening" and "narrowing", whose function is that of 'predicator'.
The bracketed element is an adjunct consisting of a coordination of two clauses functioning as a supplement, a loosely attached expression set off by intonation and punctuation, presenting supplementary non-integrated content.
It is possible for a relative clause to occur in postposed position where there is intervening material such as a supplement provided there is no confusion about the intended antecedent. Here, there is no doubt that "times" is the antecedent of "when".