How to edit grub in a persistent live drive
sudo update-grub does not work with live and persistent live systems. You need an installed system for that command line to work.
In a persistent live system, you can modify the file grub.cfg directly. If you created the system with mkusb version 12, mkusb-dus, this file can be found in partition number 3 in the USB drive.
Open a terminal window with the hotkey combination ctrl + alt + t and run the following commands to access the relevant grub.cfg. The following demo example uses Lubuntu because I have a current persistent live system for that flavour of Ubuntu, but the boot structure and the commands are the same for all flavours of Ubuntu (including standard Ubuntu).
To run a command as administrator (user "root"), use "sudo <command>".
See "man sudo_root" for details.
lubuntu@lubuntu:~$ df -h
Filesystem Size Used Avail Use% Mounted on
tmpfs 770M 1.7M 769M 1% /run
/dev/sda4 1.9G 1.9G 0 100% /cdrom
/cow 54G 102M 52G 1% /
tmpfs 3.8G 0 3.8G 0% /dev/shm
tmpfs 5.0M 4.0K 5.0M 1% /run/lock
tmpfs 4.0M 0 4.0M 0% /sys/fs/cgroup
tmpfs 3.8G 0 3.8G 0% /tmp
tmpfs 770M 80K 770M 1% /run/user/999
/dev/sda5 54G 102M 52G 1% /media/lubuntu/writable
/dev/sda1 55G 67M 55G 1% /media/lubuntu/usbdata
lubuntu@lubuntu:~$ setxkbmap se
lubuntu@lubuntu:~$ sudo mount /dev/sda3 /mnt
lubuntu@lubuntu:~$ ls -l /mnt/boot/grub/grub.cfg
-rwxr-xr-x 1 root root 2241 Dec 30 13:17 /mnt/boot/grub/grub.cfg
lubuntu@lubuntu:~$
Your live system drive (the USB drive) may be seen as /dev/sdb, /dev/sdc ... instead of /dev/sda. This is shown by the command df -h.
Since you need sudo to edit it, I suggest that you use the text mode editor nano
$ sudo nano /mnt/boot/grub/grub.cfg
Edit 1: I should add the following comment:
In a persistent live system you can add the settings, that reside in the home directory (usually as hidden files), and you can add application program packages.
But you cannot upgrade the kernel and you cannot add or modify the kernel drivers, because the kernel and its drivers are started before the overlay for persistence is activated. If that is what you want, you must use an installed system. It is possible to create an installed system in an external drive, for example a USB pendrive. Let me know if this is what you want, and I can help you with some tips and links.
Edit 2: Details to find grub.cfg
df -h indicates which device (drive letter) that is active, but does not show the partition (number) containing grub.cfg unless it is mounted.
You can use lsblk -f to show all devices including partitions (also unmounted partitions), and it helps to know what partition structure should be expected in a particular system. A system created by the tool dus-persistent (in mkusb) puts grub into partition #3.
Please ask for more details if necessary, for example if your persistent live system was made by another tool (not dus-persistent).