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How to fix "Failed to check for updates"?
Delete plutonium.exe and plutonium.new.exe and download a new copy of the launcher here: https://cdn.plutonium.pw/updater/plutonium.exe
Then place the new exe anywhere you want: on your desktop or in your game files folder for example and run it.
The update process will start and you won't have anything more to do next times, only run plutonium.exe and it'll automatically update
As for the VPN issue some servers will kick you if you use a VPN (or if they think your IP comes from a VPN). You can try to contact the server owners to get your IP whitelisted or find a solution. The fix above should fix your problem and doesn't require a VPN to work or play so you should not use it while playing Plutonium (or use split tunneling)