Hello, I'm Keith and happy to help you with your question.
It is possible that McDonalds HR department operated PAYE incorrectly. A possibility is an incorrect tax code resulting in an under deduction of tax. It is also possible that HNRC have made an error, or indeed be correct.
The first thing to do I suggest would be to calculate the actual tax due. This should be fairly simple. Take the total income from all sources including savings interest. Take away the tax deducted. This can be found for employment from the P60 [issued by the employer] or the P45 on leaving. Then from the total income deduct the Personal Allowance and then the balance remaining is taxed at presumably the basic rate of 20%. I have not quoted allowances as your question did not give the actual tax year of the employment.
From this calculation (and do not forget to take into account the moneys refunded by HMRC in your computation) you have a base with which to compare HMRC's demands which should have been received initially with a statement of the position as at the relevant year's end. Cross check this statement with your calculation and you will be in a position to dispute HMRC's demands by means of an appeal which is accomplished by merely writing a letter on the matter giving reasons. Finally, never ever deal with HMRC by telephone, there is always the possibility of somebody getting hold of the wrong end of the stick if they can even find the stick at all. Use written communication, then the approach of both sides can be pinned down exactly.
HMRC's attitude that 'We don't send letters out unless they are correct,' is a standard position, is simply balderdash and piffle! I have seen plenty of such assertions, invariably incorrect. So it is a case of check, check and double check, check and check again to either confirm or disprove HMRC's approach in this matter.
I do hope I have helped put your daughter's mind at rest with my response.