Q

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$ git remote add origin git@github.com:lut/EvolutionApp.git
fatal: remote origin already exists.

$ git push -u origin master
fatal: '  ' does not appear to be a git repository
fatal: Could not read from remote repository.

Please make sure you have the correct access rights
and the repository exists.


My keys were added succesfully

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You've successfully authenticated, but GitHub does not provide shell access.

Solve

Try and redefine the ssh url for remote origin:

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git remote set-url origin git@github.com:name/projectname.git

Only git remote set-url can change an existing remote URL (as opposed to git remote add, to add a new remote name and URL). Here, the issue was the URL of the existing remote ‘origin’, EvolutionApp: it needed to be replaced by a valid one.
Using git config url.”ssh://git@github.com/“.insteadOf https://github.com/ would not have helped, considering there was no HTTPS URL in the first place.

Finally

After input 

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ssh -T git@github.com

Although it still output 

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You've successfully authenticated, but GitHub does not provide shell access.

It just a notice .