@DaveLaQ said "The problem is that TurboTax is asking for the amount of the 1099-NEC and counting that as income"
That's because the IRS also is looking for the full amount of the 1099-NEC to show on your schedule C.
You have already identified the solution: you must show all the income received (including the reimbursement for expenses). Then, you deduct your expenses, including the ones you were reimbursed for.
Your customer has submitted the 1099-NEC correctly; they are required to include the reimbursed expenses on the 1099-NEC.
