Lets talk guns. If a semi-automatic striker fired pistol was used, there would be no 'click' involved in loading a round in to the chamber. One must 'rack' the slide on a semi-auto, making a distinct sound, a sliding 'schlick, schlick' type sound, not a 'click'. There is no trigger per se on this type of handgun.
Now, if a hammer fired semi-automatic pistol were being used, and it was not 'charged', and there was a live round in the chamber, then one would pull the trigger back to 'charge' the gun, preparing it for firing. I imagine the sound of doing this might be heard on the audio, but IMO it would be unlikely.
A revolver, with the cylinder loaded (or perhaps unloaded if that was the intent), and the hammer down....one would have to pull the trigger back to charge the gun. This type of pistol definitely makes a clicking sound when cocking it.
The whole idea of a 'click' being heard on the tape is hard for me to fathom, with the muffled sound of the voice being what it is.
So, if indeed LE has stated a 'click' is the sound of a gun being 'charged' then it would be either a hammer fired semi-automatic pistol or a revolver, the revolver being my best guess.
Note here, that a rifle or shotgun, in some cases depending on the type of weapon, would need to have the trigger pulled back to charge it for firing. But my thought is that there likely was no long gun used in this crime, and if so, the chances of it being a bolt action or semi-automatic rifle would be greater, and those type long guns for the most part don't involve pulling a trigger back to prepare them for firing. They are 'charged' for firing in different manners, some by the bolt, some by racking as in pump shotguns etc, and some by what's actually called a charging handle.
Lastly, it could be that a click might be heard, though extremely unlikely, in either a striker fired or hammer fired pistol when the safety is switched on, or off, provided the pistol is equipped with a safety. But this 'click' would be even less noticeable than the sound of charging a revolver, if even noticeable at all.
In either case, when a revolver is charged, the trigger needs very little pull pressure to cause the gun to fire. Same being with a hammer fired semi automatic. If the safety is off on a hammer fired semi automatic pistol, and the trigger is pulled backwards, the trigger at that point is extremely touchy, and will fire with a minimum of pull pressure.
Why all this? Well, it's educational, plus, IMO, IF there was a click actually heard, I'd guess it to have come from a revolver, and second to that, one of John Moses Browning's 1911 designs, or a hammer fired semi-automatic, the revolver being most likely to make a noticeable noise when charging the trigger.