sudo apt-get update failing - "could not open" list file due to "permission denied"

The contents of that directory are partially downloaded lists (perhaps APT was interrupted?) which can be safely removed.

Start a root shell, being aware that you now have permission to do all kinds of damage to your system, so be careful (we need to do this because sudo rm errors out)

sudo -i

Remove the contents of the directory (type carefully) and exit the root shell:

rm /var/lib/apt/lists/partial/*
exit  

Then run an APT update to regenerate the package lists and check the problem is gone:

sudo apt update

Solution nearest to @Zanna answer But include save old APT lists folder as backup first before delete partial file from APT lists folder.

First clean APT

sudo apt-get clean

Make backup APT list folder

sudo cp -r /var/lib/apt/lists /var/lib/apt/lists.old

Remove partial folder from APT List folder

sudo rm -rf /var/lib/apt/lists/partial/

Then clean and update APT

sudo apt-get clean
sudo apt-get update

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