In my home directory I have files in a local git repository, because I want track them all under version control.
Most of these files I want to push to a remote repo
This is not possible. If you have committed a file, then it will be pushed. The push action only acts on commits, not on a file basis.
But you could use a submodule for your sensitive data and keep this submodule on your local machine, while you push the regular git repository to the remote machine.