Can I include an additional charge after the invoice has been generated?

By default, you cannot add a charge to the same invoice after it is generated. But you can raise a separate invoice for the additional charge using the Add Charge option (on the right panel) on the subscription details page. The same option on the Customer details page will allow you to add a coupon.

However, if your business involves usage-based charges, you can enable the Metered billing feature on your site. This will generate every renewal invoice in PENDING state and you can add extra charges to the same invoice. More information here.