Yes, you simply indicate the income made in the Business Income and Expenses section of TurboTax along with the associated business expense.
The standard deduction is for your personal deductions, not your business expenses. You need to enter the business expenses to show a business loss.
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Don't confuse itemized deductions on schedule A with your business expenses/deductions on schedule C. They are separate. For Schedule A personal deductions, you get to take your itemized deductions or the standard deduction, whichever is larger. Itemized deductions are things like Medical, Gifts to Charity, State Income Taxes Paid, Mortgage Interest, Property Taxes, Car Registration fees, etc.
You get to take both, your business expenses AND the Standard Deduction (or your personal Itemized Deductions). And you have to enter your business expenses. Be aware, if you have self employment income you can get in trouble for not reporting all your expenses to qualify for the EIC.
You can enter 1099Misc or cash Self Employment Income into Online Deluxe or Premier but if you have any expenses you will have to upgrade to the Self Employed version.
How to enter self employment expenses