The only way she can claim the mileage would be if she paid you for the mileage, then you would need to count this payment as income and report it as income on your return. If she is paying for gas or tolls, then she can claim those expenses, but for actual mileage, she needs to own the vehicle.
It may be a good idea to open an account for your child with a card attached so you can pay the expenses such as hotel rooms and fuel costs directly from her account. That way you have all expenses and income in one account to make it easier for tracking purposes and the waters don't get muddied.
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