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Linux Mint tells me, I only have 622 MB free disk space but there should be some gigabytes left.

Looking at the partitions I am told that there are about ten gigabytes unused. I googled the problem and didn't find a solution but I did find the hint that I should check the disk usage with df -h.

sudo df -h /home
Filesystem Size Used Avail Use% Mounted on
/dev/nvme0n1p8 189G 178G 622M 100% /home

The output doesn't make any sense to me: The difference between Size and Used is 11GB, but it only shows 622M as Available.

The SSD isn't old, so I wouldn't expect such a discrepancy.

What should I do?

If the filesystem is ext4, there are reserved blocks, mostly to help handling and help avoid fragmentation and available only to the root user. For this setting, it can be changed live using tune2fs (not all settings can be handled like this when the filesystem is mounted):

-m reserved-blocks-percentage

Set the percentage of the filesystem which may only be allocated by privileged processes. Reserving some number of filesystem blocks
for use by privileged processes is done to avoid filesystem
fragmentation, and to allow system daemons, such as syslogd(8), to
continue to function correctly after non-privileged processes are
prevented from writing to the filesystem. Normally, the default
percentage of reserved blocks is 5%.

So if you want to lower the reservation to 1% (~ 2GB) thus getting access to ~ 8GB of no more reserved space, you can do this:

sudo tune2fs -m 1 /dev/nvme0n1p8

Note: the -m option actually accepts a decimal number as parameter. You can use -m 0.1 to reserve only about ~200MB (and access most of those previously unavailable 10GB). You can also use the -r option instead to reserve directly by blocks. It's probably not advised to have 0 reserved blocks.

Deleted files can also contribute to "missing space"

lsof | grep deleted | grep /home

returns this output for me

chrome 11181 criggie 15u REG 254,0 
4194304 50651663 /home/criggie/.config/google-chrome/BrowserMetrics/BrowserMetrics-5D0236AF-2BAD.pma (deleted)

Which shows that Chrome running as PID 11181 opened that BrowserMetrics file then deleted it, and still has the filehandle open. This means the file is invisible in a directory listing, but is still taking up disk space.

Why do programmes do this? When the running binary terminates, the OS will release the open file handle and the file on disk will be gone, without risk of leaving a stale temp-file around.

What I can't see is how big that file's disk usage is.