See the autocheck parameter, which turns all HTTP errors into fatal errors. When you turn autocheck off, you will need to do all error checking yourself.

my $mech = WWW::Mechanize->new( autocheck => 0 );
my $res = $mech->get('does.notexist.example');
$res->is_success
    or print "Uhoh\n";
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Alternatively, use eval to trap fatal errors when making the original connection.

my $mech = WWW::Mechanize->new();
my $connected = eval {
    $mech->get('does.notexist.example');
    1
};
if (! $connected) {
    print "Uhoh\n";
};
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Also see Try::Tiny to reduce the exception-dance to one less step (I haven't used it myself):

use Try::Tiny;

my $mech = WWW::Mechanize->new();
try {
    $mech->get('does.notexist.example');
} catch {
    print "Uhoh: $_\n";
};
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