How do I stop firefox icon reappearing on the desktop?

It sounds odd, and it appears someone has already submitted this question with no resolution about a year ago, but an icon for Firefox keeps appearing on my desktop. It was originally the actual firefox icon, but now it appears to be an 'installer' type icon. Is honestly spontaneously appears when I boot up, and reappears at random times, regardless of my efforts to prevent it from doing so.

I have reinstalled (originally keeping preferences, then fully clean install using Revo Uninstaller). I am on 18.0, and I see there is an update waiting for me to restart.

I have worked in IT and am fairly sure I am not a complete idiot, but this is truly annoying.

Please help!

P.s. Interestingly, when this started (about release 17 I think - earlier in 2013), there was an issue with the automatic updating - every time I tried to start FF, it insisted on me 'rebooting to complete a previous install'. I eventually fixed that issue (obviously rebooting didn't fix it).

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