Escapes within double quotes can be tricky. I suspect what you want is one of the following:

$regex = "href\\s*=\\s*\".*?\"";
$regex = 'href\s*=\s*".*?"';
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It might help you to print $regex and $match, to see what you are getting.

update: You can use $match in the substitution (at least it worked fine in my tests) but it won't do what you expect if $regex isn't what you expect. Also, since $match is one particular match, adding the global qualifier to the substitution using $match isn't likely to replace multiple instances.

You have indicated that you have more to do than just substitute the href values in the anchors. Without knowing more about what you are doing it is hard to make an appropriate recommendation but something like the following might be closer to what you want:

use strict;
use warnings qw( FATAL all );

my $regex = 'href\s*=\s*".*?"';
print "regex = $regex\n";
foreach my $line (<DATA>) {
    chomp($line);
    while ($line =~ /$regex/) {
        my $match = $&;
        $line =~ s/$match/xxx/;
    }
}
__DATA__
No anchors here
<a href="asdf">go there</a> <a href="fdsa">go here</a>
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