When you are going through the program, it will ask about additional expenses that must be paid to the college. Enter the additional $5,000 there. You can enter books and other supplies in the box below.
See Qualified Education Expenses - Internal Revenue Service for available expenses.
The 1098-T is only an informational document. The numbers on it are not required to be entered onto your tax return. However receipt of a 1098-T frequently means you are either eligible for a tuition credit or possibly your student has taxable scholarship income.
If you claim the tuition credit, you do need to report that you got one or that you qualify for an exception (the TurboTax interview will handle this)
You claim the tuition credit, or report scholarship income, based on your own financial records, not the 1098-T. In the 1098-T screen, click on the link "What if this is not what I paid the school" underneath box 1. You will then be able to enter the actual amounts paid.
Or if you find it easier, just change the numbers in boxes 1& 5 to what your records show. The 1098-T that you enter in TT is not sent to the IRS.
Be sure the $5K extra you paid was tuition or other qualified expenses. Room and board and some activity fees are not qualified expenses for the tuition credit and/or tax free scholarship.