your permissions indicate that only root has write access. As all TLJH users are added to a
jupyterhub-users group (https://tljh.jupyter.org/en/latest/topic/security.html#system-user-accounts), you could change the drive’s group owner to
jupyterhub-users with something like:
sudo chown -R root:jupyter-users /mnt/my_mounted_drive/
(of course you need to adjust the path to your drive), and then give the group write permission:
sudo chmod -R g=rwx /mnt/my_mounted_drive/
And one word of caution: in your example you are sharing a notebook. As all users access the same file, it will create an absolute mess when they all open and modify that notebook. This is why usually users don’t get write access on such a shared location.