Hi Nina, it's easy - there's a Sage guide here that explains it all.  

Basically, there are 2 methods and you do NOT issue another P45 in either but you must send them a letter or payslip to show the extra payment.

1) Rollback the final pay and process it again using the same tax code as before and issue a revised payslip and send an FPS adjustment.  (Personally, not my preference - messes my reports up!)

2) Process the extra pay in the next pay-run depending on whether it is a regular (final salary/wage) or irregular (accrued holidaypay/bonus/backpay) payment - both require use of a 0T tax code on a W1/M1 basis but if it is an irregular payment and the employee isn't paid WEEKLY you will need to calculate their NI manually using the weekly thresholds - the guide explains it all. 

Should the employee question not sending them another P45 (some do!), you can link them to this HMRC page to scroll to the bottom for the "paying an employee after giving them a P45" part and the final line in the section which explicity states you must not give them another P45.  

EDITED: To amend reference to monthly in option 2 to the correct value of weekly!