In none of Adagio Receivables, Payables or Invoices can you edit a posted transaction to change field information, including the Job number. To make corrections such as these, it is usually required to reverse the transaction by adjusting it out and then posting the correct transaction.

That the invoice is paid makes this more complicated. The invoice number is saved with the payment, and this cannot be reversed. I believe you would have to reverse the payment in addition to the invoice, and re-enter both. Try this on a backup copy of your data before doing it live.

Also, there is the issue with the information retrieved into Job Cost (which you shouldn't post if it's not right). I think you can just delete the incorrect entry, as it will be replaced by the newly posted corrected entry.