I have three of the TSW series pistols that I carry for duty. A 5906TSW for summer carry, a 4046TSW for fall/spring and a 4583TSW for winter carry. I've noticed that on all three that not only is the reset quite long, there are two noticeable "clicks" before the trigger resets. I have other guns and that second click isn't there. My SIG's, HK's amd 1911's don't do it. I shoot the smiths just fine and love em thats why I carry them exclusively. I've gotten used to when I feel the click on the return I pull the trigger again but on the Smith's it's taking more than some getting used to.

So my question is: is this normal or not and is there a way to eliminate the first click?

It's been brought up before.
A search for "trigger Click should bring it up.

Is this anything to do with the trigger reset spring being broken?

This is a common issue with 3rd gens. The trigger "play" spring is either broken or lost tension.

My 4566 has the same issue. Mine is not broken, but won't hold tension. I adjust it with a pencil per the recomendations of other threads (and S&W CS itself) but it only works about one mag before clicking again. I probably should quit messing around and have it replaced.

The up side is that it doesn't affect trigger operation. It's just going to be annoying in SA mode.

If you remove the slide, the spring (actually a small brass-colored leaf spring) is located at the rear of the trigger assem. just forward of the mag-well frame. Press it forward (a pencil is the commonly mentioned tool) and that should clear up the click, for a while at least.