I started getting a 2nd shortcut icon on my desktop for Mozilla Firefox. Every time I delete this 2nd icon, it comes back and I don't know why.
Sometimes a problem with Firefox may be a result of malware installed on your computer, that you may not be aware of.
You can try these free programs to scan for malware, which work with your existing antivirus software:
Microsoft Security Essentials is a good permanent antivirus for Windows 7/Vista/XP if you don't already have one.
Further information can be found in the Troubleshoot Firefox issues caused by malware article.
Did this fix your problems? Please report back to us!
Which security software (firewall, anti-virus) do you have?
Try to boot the computer in Windows Safe mode with network support (press F8 on the boot screen) as a test to see if that helps to remove this second shortcut.