> HMRC does sometimes make mistakes with people's tax codes, resulting in > the wrong amount being taken off your pay, and your employer doesn't > (and can't) correct those mistakes for you

That is correct. However, HMRC sends you a letter with every tax code change, which is calculated and posted automatically; I've once received two separate letters, with two different tax codes, sent literally one day apart from each other. :D

And if you feel that you tax code is incorrect it is very easy to check your online HNRC account and correct any information they might have.