It’s easy to see why proofreading at the earliest stages of a project is so important: The earlier a mistake is discovered the less it costs to fix. Last minute changes on press can be very expensive.
Standard industry practice is for customers to bear responsibility for content they have approved. Changes made after approval of a proof, or that change the scope of work after files have been preflighted, are either printer’s errors (PEs) or author’s alterations (AAs – from the days when printing meant books).
Printer’s error means your printer failed to match the approved proof, and they must cover the cost for any corrective work. AAs means you created new work, and you must cover any additional costs to perform it.