Commuting from home to work and from work to home is commuting, and is not deductible on your tax return. if you are doing other driving around during your work day, that can be entered on your Schedule C as a business expense.
https://ttlc.intuit.com/questions/1901340-where-do-i-enter-schedule-c
https://ttlc.intuit.com/questions/3398950-what-self-employed-expenses-can-i-deduct
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https://ttlc.intuit.com/questions/2893392-where-do-i-enter-a-1099-misc
https://ttlc.intuit.com/questions/2903027-how-do-i-report-income-from-self-employment
Although it may be news to you that you “have a business” (and most of the other people who get their first 1099-MISC); The IRS considers anything in box 7 of 1099-MISC to be self employment income. Enter the 1099-Misc income, at Other Common income / income from a 1099-MISC . TurboTax (TT) will complete Schedule C or C-EZ for you and allow you to deduct any expenses associated with this income. You'll also have to pay self employment tax (social security & Medicare) on any profits greater than $400. These types of payments are frequently done this way. Your "employer" is not the only one.
As others have stated, commuting is not deductible, for either the self employed or regular W-2 employees.