perform a regex on $r to extract the url of the image

That's fragile and error-prone. Much better to use WWW::Mechanize's built-in methods, since they actually parse the returned HTML:

$mech->get($add);
my $img_obj = $mech->find_image( url_regex => qr{captcha\.php} );
$mech->get( $img_obj->url, ':content_file' => 'captchar.jpg' );
[download]

... or somesuch

Update: sorry, somehow (!) didn't see Anonymonk's post