Shortly before you were due to start receiving your state pension, you should have received a P161
to complete and send back to the tax office to help them ensure that the correct amount of tax was deducted from your ECC pension and that your tax code took into account your state pension which is paid gross but is taxable.
When you started the part-time job, the employer should have asked you to complete a P46 in which you would have confirmed you were in receipt of your ECC pension. The employer would then have submitted that to the tax office for coding instructions. That doesn't appear to have happened. HMRC would have picked up your ECC pension from your NI number on the P46 and issued an appropriate tax code, probably BR (basic rate tax deduction - 20%).
The law states that a taxpayer is responsible for ensuring that their own tax affairs are in order and the tax office have to accept that it was reasonable for them to believe that they were in order for them to be absolved of any responsibility for an underpayment of tax accruing, leaving an employer, a pension payer or HMRC to carry the can. I always tell people who take on a second job to call the tax office themselves to have their tax coding(s) looked at in the light of a second PAYE source rather than trust others to do it.
The first thing you need to do is to satisfy yourself that the tax office P800 calculations of the underpayment(s) is/are correct. If they are, you should write to the tax office and ask them to review your case, give you an explanation as to why you are underpaid and to write off the tax owing under Extra Statutory Concession A19 which they should do if they did not use information supplied to them that should have enabled them to get your taxes correct within the time limit set out in ESC A19.
Take a look here
for information on ESC A19 and on employer error and ask the tax office to look into your part-time employer's actions when they took you on. Also refer to the number of letters you have had, all giving you different figures. If you have not had P800 tax calculations, call the tax office and ask for them so that you can check the figures. There are some more useful notes here
If the tax office maintain they weren't responsible for the underpayments after reviewing your case, you should ask for a second review by another tax officer. After that, if you need to, you can make a complaint to The Adjudicator's Office
and then to The Parliamentary Ombudsman
I'm afraid the process I've outlined above is the one you to follow to have any chance of having the tax owing written off. There is no other way to deal with the situation you find yourself in.
If you do have to pay the tax you can negotiate with the tax office to pay it over a period of time.
I hope this helps but let me know if you have any further questions.