rm "~/Desktop/$filename"
rm "$HOME/Desktop/$filename"
Tilde does not expand to the user's home directory when it's single or double quoted. Use double quotes and $HOME instead.
$HOME
Alternatively, the ~/ can be left unquoted, as in rm ~/"Desktop/$filename".
~/
rm ~/"Desktop/$filename"
If you don't want the tilde to be expanded, you can ignore this message.
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