Hi,
If my customer cancels an order...
How does your customer cancels an order ?
If you're talking about the "cancel" button of the orders listing, then yes there is an email sent.
If you don't want that, you would have to edit the file components/com_hikashop/controllers/order.php add add such line:
$newOrder->history->history_notified = 0;
before the line:
$orderClass->save($newOrder);
You can't add complex conditions like "only if they did not also make a purchase within a half hour or so." unless you add some more complex custom coding.