Full order cancellations can be triggered from Brightpearl as an order status change. The cancelled order status in Brightpearl should be mapped to the cancelled order status in BigCommerce.
Partial cancellations, if a customer wants to remove a specific item from an order there are a few different approaches that can be taken. BigCommerce does not support, line item refunds/cancellations so an order would need to be cancelled and a new order placed. One other option is updating the order directly in Brightpearl and then granting a credit on BigCommerce or refunding the partial amount using Brightpearl Payments.