Looks like you've been unlucky as the amount is at the upper end of the 'average' scale.
In some cases, HMRC will consider writing off demands where taxpayers can demonstrate that they provided all the information necessary to calculate their tax correctly. You can dispute extra tax charges by claiming on a ESC19 form proving that you have supplied information in good faith and retrospective bills should be dropped, but it depends on the reason for the underpayment.
You first need to check that the HMRC's new calculation matches the information on your P60 for the particular year.
if it can be proven thet the HMRC did not act in a timely fashion, then you can apply for an Extra Statutory Concession via the aforementioned form.
The Low Income Tax reform group have been fighting the 'little mans' corner and there may be more info on their website:
http://www.litrg.org.uk/