There are several points here.
First, the agency that pays you can choose to reimburse any expense, or not, at any rate. That's up to your contract with them. There are two ways to do this and its not clear what method you are using.
A. You submit receipts, they reimburse but don't include the reimbursement on your 1099-NEC, and you don't include the reimbursement as taxable income and don't report the expenses as a business expense deduction. With this method, you must still include as taxable income, any reimbursement that is more than your qualified business expenses.
B. The agency pays you for your car expenses (with or without receipts) and does include the payment in your 1099-NEC compensation. In this case, you include the income and deduct your actual business expenses on schedule C. If your expenses are more than the reimbursement, you get an extra business deduction, and if your expenses are less than the reimbursement, the difference is part of your taxable income.
Second, regarding a car that you own:
The agency can reimburse you at any rate they want. If they reimburse less than the IRS rate, you can claim a deduction for the difference. If they reimburse at more than the IRS rate, the difference is taxable income. How you report this depends on whether or not the agency includes the reimbursement in your 1099-NEC compensation.
Third, regarding a car you rent:
You can use the standard mileage rate or the actual rental cost**. Here, you must keep mileage records, and if you use the rental (let's say) 80% for business, then 80% of the cost is a business expense. How you report this depends on the agency. For example, the agency can reimburse you for the actual cost, leave it off the 1099, and then you don't report the expense on schedule C. Or, the agency can reimburse you and include the reimbursement on your 1099, and then you deduct the expense against the income on schedule C. If the reimbursement is more than the expense, the difference is taxable, and if the reimbursement is less than the expense, you take the additional deduction.
However,
If you rent or lease a car for more than 30 days, and the fair market value is more than a set threshold ($56,000 for 2022), you must also include a portion of the fair market value of the car in your taxable income. (You don't actually include the amount as taxable income, but you reduce the expense deduction.) This is described in chapter 4 of IRS publication 463. I don't know if Turbotax includes this calculation because I have never tested it. I reviewed the discussion in publication 463 and I don't understand how the calculation works, so if you plan to rent a car more than 30 days, you may want to consult a tax professional.
https://www.irs.gov/publications/p463#en_US_2022_publink100034047
**Edited to add. In one place in publication 463, it says use actual rental expenses for a rental car. In another place, it says that for rentals or leases, you can use either the actual expense method or the standard mileage method. The IRS seems to treat leases and rentals as the same thing, provided the period is more than 30 days. You may want to check with a tax professional.
