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use strict;
use warnings;
...
$string =~ s/”//g;

dd $string;

C:/Strawberry/perl/bin\perl.exe -w d:/tmp/pm/replace_quotes.pl 
"copied from a web page looking like this: \xE2\x80\x9CTest.\xE2\x80\x
+9D "
"copied from a web page looking like this: Test. "

C:/Strawberry/perl/bin\perl.exe -w d:/tmp/pm/replace_quotes.pl 
"copied from a web page looking like this: \x{201C}Test.\x{201D} "
"copied from a web page looking like this: Test. "