The trigger play spring, like the name suggests, stops the trigger from moving rearward without any force on the trigger face when the hammer is captured by the single-action notch. It fundamentally provides some tension before the actual trigger engagement occurs. If is is broken, bent, or absent, the trigger will flop around, and it can make a clicking sound. The sound is basically the same if the spring is there but bent or if the spring is broken.
The click really has nothing to do with the spring. The click is usually caused by a condition where the front hooks of the trigger's 'ears', or prongs, have slipped up and forward over the top of the drawbar head's V-notch (in which the prongs normally sit to some degree). The clicking noise is usually heard as the tips of the prongs are being snapped back down over the edge of the drawbar's head at the top of the notch. The large frame (10xx and 45xx), like your sample, are more prone to the issue. [Borrowed from "Fastbolt", a S&W armorer for a CA LE agency at
http://smith-wessonforum.com/s-w-smi...ay-spring.html ]
Brownells has the spring at
https://www.brownells.com/handgun-pa...prod14780.aspx and the rivet at
https://www.brownells.com/handgun-pa...prod14429.aspx . The rivet is not in stock at Brownells. I did not check Numrich or MidwayUSA.
I would take the spring out as broken pieces of spring can tie up the pistol. It works fine, as noted, without it. But it can make a clicking sound as the trigger moves around without tension unless one replaces the trigger play spring and rivet.