Yes, your employee's wages are deductible.
Because you are providing both a truck and driver, it sounds like you aren't really "leasing" the truck out, but the Company has hired your business as an Independent Contractor for Trucking.
This is a business that will go on Schedule C (assuming you have not incorporated your business), and "wages" is one of the items it will ask for.
It sounds like the driver is probably your employee, so you will need to file quarterly payroll returns, file a W-2/W-3, Federal Unemployment, State Unemployment, Worker's Compensation, and possibly other State requirements. TurboTax does not do any of those items. You want to can hire out payroll or use Quickbooks for payroll.
As your first year of business, I would highly recommend a tax professional for at least the first year.
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