If you've created a discount set in Bulk Discounts and only want to delete specific codes from that set, you can do it through Shopify's Admin.
1. Click Discounts on the left of the Shopify Admin
2. Select your discount set - Any sets created through an app will say so next to the name
3. Click View all codes
4. Select the discounts to delete by checking the box next to it
5. Click Bulk actions > Delete codes
6. Confirm the action when prompted