Use the --git-dir and --work-tree option of the git command:
If you are modifying files in my_project, but want them taken into account in my-project-repo git repo, you can do:
git --work-tree=/www/my-project/ --git-dir= /github/my-project-repo/.git status
git --work-tree=/www/my-project/ --git-dir= /github/my-project-repo/.git add -A .
git --work-tree=/www/my-project/ --git-dir= /github/my-project-repo/.git commit -m "add files from my-project"
You might want to refresh your working tree in /github/my-project-repo after modifying its index with your git add.
cd /github/my-project-repo
git checkout .
Beware though of concurrent modifications you could have on common files: that last checkout would erase and overwrite them by what you added from /www/my-project.
If you want to be sure to preserve any work in progress in /github/my-project-repo:
 git stash save --keep-index
 git stash drop
The OP found a simpler solution, and:
- moved the storage directory to /www/.
- cd /my-project/,
- git config,- git init, etc.
I stop tracking the repo that was save on Docs/Github/ — my first storage directory.