Usually when you want to suppress a single icon from showing up in the System Tray, you right-click on the Taskbar, select Properties, and in the Taskbar tab select "Hide inactive icons" and click Customize. The Customize window lists items that you can selectively suppress, but I have found that if the name of the icon changes, the icon isn't suppressed consistently. However, choosing "Hide inactive icons" tends to make most of the...
icons in the System Tray hide themselves.
As for the icon on your desktop reappearing when it's deleted, that will depend on what it is. If it's something that has been defined by an administrator as being part of a Group Policy action, for instance, you won't be able to get rid of it unless the Group Policy is changed. If it's an icon that's being put there by a malware application (such as a "Get XXX Free!"-type marketing icon), then you'd need to do a spyware sweep to make sure you aren't infected with anything like that.
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