I know what the solution is. I'm not familiar enough with the plugin to know what exactly is going wrong, but here's my guess:
You have your "slide" <div> hidden from view, ready and waiting.
You turn that "slide" <div> into an image cycle.
When the user clicks the link, you're copying the cycle's HTML into a different <div>.
The original remains an image cycle, but the new one is still just markup.
The solution is to turn the new markup into an image cycle instead of the original. Don't set up the cycle at the very beginning of your script, do it after you copy the HTML over. (Also, once that happens, you'll need to prevent the default action from the "Item 1" link.)