Can someone please tell me what the function of this module, and more specifically, the curley wires on top of this contactor? It looks like the wires go to the same terminals as the main contactor terminals (although I didn't test continuity to confirm), and from what I can tell, it engages at the same time as the contactor via connecting to the "buttons" on top. I tried to search Google but not knowing what to call it made that futile. The main reason I am asking is that the small curley wires are extremely thin, almost as if it served as some sort of a fuse. Any help would be appreciated.
I think you have omitted to mention that these contactors are part of a power-factor control panel and so they are switching a purely capacitive load.
If so, those coils are inductors.
Figure 1. Power-factor correction capacitor soft-start.
Remember that a discharged capacitor appears as a short-circuit when connected to a voltage source. A huge inrush current occurs and this is hard on the contacts.
The auxiliary contact block on the contactors is designed to close before the main contacts. This allows a charging current to flow into the capacitors to partially charge them before the main contacts make and reduces wear and tear on the them. When I first saw them on an installation I was very surprised that such fine wires could do that work.
According to a couple of online calculators including MustCalculate the inductance for my estimate of Ø = 10 mm, L = 40 mm and 9 turns the inductance is about 180 nH. This has an impedance of only 56 μΩ at 50 Hz but it's effectively a transient that's being switched so the impedance will be much higher.