How on earth could a cat do that? Sorry, that got me laughing.
Anyway, if you can't find it in the Recycle Bin, I'd suggest looking into Start menu or go into Program Files/Origin Games/Fun With Pets and somewhere you should find the launcher file (I can't say where exactly it is: my computer deleted the whole SP8 folder while having another "error tantrum").
Deleting the icon on the desktop shouldn't uninstall the game (as far as I know).
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I would first check the Recycle Bin.
Anyway, if you can't find it in the Recycle Bin, I'd suggest looking into Start menu or go into Program Files/Origin Games/Fun With Pets and somewhere you should find the launcher file (I can't say where exactly it is: my computer deleted the whole SP8 folder while having another "error tantrum").
Deleting the icon on the desktop shouldn't uninstall the game (as far as I know).