I am sorry to hear about your partners illness and glad that he is on the mend.
The first �30k of any redundancy payments are tax free. If the payment was in excess of this, the extra tax would have been paid through PAYE. This would be on his P60. If his P60 has been destroyed, I am not sure if his old company can/will provide a copy or evidence of earnings/tax. If the HMRC claim that they have no evidence of this, this will need to be taken up with his old company. I am struggling to see why they would need this if these events were in different tax years, unless he had a Gap between being redundant and starting his new role, as well as the gap when he was ill. I had a gap when i was made redundant the last time and made the mistake of not claiming benefits - HMRC took a long time to be convinced that I was neither earning or claiming benefits for 4 months. I hope that you get it sorted out.