For what period do they say you owe the £5,300? If it is for the year to 5th April 2008 then it would seem that you did not query your code that was being operated during that year and this included relief for pension contribs that you were not entitled to.
Therefore I would have in those circumstances to disagree that it was their fault - unless of course you had written asking them to remove any such allowances from your code.
There are provisions for challenging items which arise as a result of HMRC error but the provisions are quite stringently applied and include the condition that you would have had to be of the view that your tax affairs were in order and that you had done everyhting correctly. If therefore they could show that you accepted incorrect codings then they would not apply the Extra Statutory concession. You will find the HMRC Guidance to their staff in relation to the concession at the following link http://www.hmrc.gov.uk/manuals/pommanual/PAYE95025.htm
and there is also a flow chart at this link http://www.hmrc.gov.uk/manuals/pommanual/PAYE95085.htm
As far as the current years code is concerned I assume (IF IT IS AN INCORRECT ASSUMPTION no doubt you will let me know) that they have changed the code on a cumulative basis (that effectively recovers unpaid tax that would have arisen from 6th April to the date of change) as opposed to reducing it on a week1/month1 basis (effectively does not recover tax unpaid from previous 6th April to date of change but charges only the correct tax for the remainder of the tax year - leaving the unpaid tax to be recovered in future periods). If your employer has already operated the code on a cumulative basis it is probably a bit late in trying to get it adjusted but you could always try a phone call to HMRC
Tax Preparer
Fully Trained Inspector of Taxes / District Inspector/ Team Leader/ Offshore Consultant/ 41 years