How do I add Photoshop Elements icon to desktop to open to Elements screen?

same as what I did. Open file explorer and navigate to where PSE is installed, scroll down and look for PSE Editor application, and copy that to desktop.

Your question is not entirely clear.  If you want to always open the Editor when you click on the existing desktop icon, do the following:

1.   On the Welcome Screen that probably appears, click on the cogwheel icon
2. From the dropdown menu, choose Photo Editor.
3. Click Done.

Now, when you click the desktop icon, the Editor will always open.

Let us know if I have misunderstood your question.