That's a connection time-out. Your webserver may have been trying to do a verify_peer, and timed out.. GoDaddy seems to have a problem with that.
Try changing contact-inc.php
$verifyResponse = file_get_contents('https://www.google.com/recaptcha/api/siteverify?secret=' .
$recaptchaSecretKey.'&response='.$_POST['g-recaptcha-response']);
to
$STRopts = array(
'ssl'=>array(
'method'=>"GET",
'protocol_version' => 1.1,
'verify_peer' => false,
'header'=>"Cache-Control: no-cache, must-revalidate\r\n" .
"Cache-control: max-age=0\r\n" .
"Connection: close\r\n" .
"User-agent: Mozilla/5.0 (contact-inc.php - saratoga-weather.org)\r\n" .
"Accept: application/json\r\n"
)
);
$STRcontext = stream_context_create($STRopts);
$verifyResponse = file_get_contents('https://www.google.com/recaptcha/api/siteverify?secret=' .
$recaptchaSecretKey.'&response='.$_POST['g-recaptcha-response'],false,$STRcontext);