Additional Cleanup

Once the payment is reversed, there are a couple of additional areas to clean up.

For Canceled Event Registrations/Invoices

Invoice record: Update the related Invoice record (Invoice Total field and Invoice Status field) and the Invoice Detail record (Line Total field) to "0.00" and "Cancelled." This ensures that they won't get confused if they look at their purchase history online, as it won't look like they owe what they were just refunded.
Event Participant record: If the refund was because the person is no longer attending an Event they registered for, you might also want to change their Event Participant Status to 05 Cancelled.
Process the Payment as per your usual process. The Payment will need to be marked with Processed set to Yes for the correction to properly show online.

Update the Invoice to have the correct, adjusted Invoice Total (and invoice Status, if needed).
Keep the Invoice Detail with the original total.
Add a new Invoice Detail record to the Invoice for the amount of the refund/discount. This will probably need to be a negative amount.
Create a Payment and Payment detail for the refund/discount.
The Payment Detail for the refund/discount should be updated to connect to the new Invoice Detail.