Voided check cashed -- how to clear open invoice?

We voided a check that a customer told us they had lost, and then they cashed it :-(

I found instructions in the Knowledge base for dealing with the situation but it doesn't make sense to me:

"So as a workaround, enter a new payment with the same data of the voided payment, pay the invoice again, and ensure that the accounting of the new payment is in the same period as the voided payment, so that GL data is not affected."

If we're paying the invoice again, won't this create another check?  How to we "pay" the invoice in the A/P system without issuing another check?