Sync git private repo with that of cloned public

We are using feature branching on a centralized master branch. That master branch is a git clone from another public repo. We just remove the .git and git init ourselfves.

Now, we need to update our own master to sync with the "parent source repo". We can manually git clone the public and copy to our own private repo. But im not sure thats the most efficient way.

How would one do it better when we need to update the private master with that of the public?