I had the same problem of not having permission to write to '/backup' - I finally figured out that it was expecting to find a folder within the backup folder with a name matching the key file's name. E.g. for a key named "id_borg.pub", I had to configure the client (in my case, Nextcloud AIO) to backup to the folder '/backup/id_borg.pub". I don't have any experience with other clients, but I hope this helps!