If you are sending any letter which contains anything such as an admission or an offer, always mark the letter without prejudice save as to costs. The effect of that little phrase is that they cannot produce it in court as any evidence of guilt or admission or that you had previously said that you would settle for an amount.
You need to tell the company in writing that they are mistaken with their facts and give them the correct facts with regard to the timing.
Tell them that you are willing to send them the payment provided they withdraw the incorrect allegations and that any future reference supplied to a prospective future employer of yours does not contain any reference to this issue.
You can also ask them to confirm where, in any employment contract which you agreed to, and which you signed it says that they are entitled to any recoupment for any training undertaken while you are employed
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