As a hairstylist, I use Groupon to promote my business and entice new clientele. Can I write off this loss revenue?

You can enter this advertising cost in TurboTax using the instructions below:

• Go to Federal > Income & Expenses
• Under Your income and expenses, click Edit next to Self-employment income and expenses
  work
• On the next screen, click Edit next to your Line of work
• Scroll down to the Expenses section and click Add expenses for work
• On the Tell us about any expenses for work page, scroll down and expand Less Common Expenses and select Advertising and then Continue.
• You can enter Groupon in the description and then the amount.
  

What's the difference between expenses and "deductions and credits"? Both ask for home office, travel, etc. After inputting expenses from Groupon losses as advertisement, I'm showing a 9k loss for the year. Is this normal? Will it be a red flag for an IRS audit?

Expenses are actual business expenses incurred for your business.  
Tax credits reduce the amount of income tax you owe to the federal and state governments. Tax deductions allow individuals and companies to subtract certain expenses from their taxable income, which reduces their overall tax bill.
As long as you have documentation to proving the expenses that you show, you will be okay.

Deductions & Credits home office is only for employee job expenses for W2 income not schedule C self employment business expenses.

Do you pay for the Groupon advertising?  Because you can NOT deduct the discounts you gave the clients unless you first included it in your income.

You can't deduct income you never received.  You will just have less income to be taxed on.  You take the loss by still deducting all your expenses but not your time or labor.  Unless you entered the full amount before the discount as income, then you can deduct the discount under Advertising or Misc Expense.

Groupon advertising... so, if Groupon paid me net revenue of $10,000 and they took $5000 off the top (cumulative) as their commission, the $5000 isn't a legitimate advertising expense that I can post as an expense?

Depends.  If you included the full gross amount before commissions as income then you can expense the commission. Are you saying the gross income was 15,000 and they kept 5,000 giving you a net of 10,000?

Can I write off this loss revenue?

What lost revenue? I'm not clear. I get the impression that assumptions are being made that may or may not be correct. What kind of tax reporting document did Groupon send you? A 1099-K1? A 1099-MISC? Something else? On whatever form they sent, in what box is the income reported? Is the amount reported "only" the amount that you actually received from Groupon?

Groupon does not send 1099s or any other tax forms.  They provide a merchant agreement with a "Remittance" amount per voucher they sell on your behalf.  For instance, if your retail price is $600 for a service, they will sell it for $300 and then multiply the total number of vouchers sold for the most recent 2 week period and pay you 65% of the gross amount they charge.  For instance, if 10 vouchers are sold on their platform @ $300 each, they'd generate $3000 for themselves, then send you a one-time payment to your checking account for $1950.  The cost to generate that $1950 is the commission paid to Groupon of $1050.  There is no invoice for the $300 per voucher on your side, they simply skim it off the top and pay you the difference.  But it's a legitimate expense to generate the remittance amount.  You never see the $3000, only Groupon does.

No.  They do a sweep every 2 weeks.  They take the gross amount for themselves and pay the merchant X% of the gross as their "Net".  For instance, a service costs $200, they sell it for $100 and pay the merchant a 'Remittance" of $65.  It's a terrible deal (which is why I don't do it anymore) but am not seeing why the $35 that Groupon takes off the top wouldn't be considered a legitimate commission, as it's the amount of money that I paid to them to make the sale.  I had to pay them $35 to generate $65 in gross revenue to me for that sale.   

@woodydog  - You report $65 as gross revenue (income), because that's what you received.  You didn't pay Groupon anything. 

Now, had Groupon issued you a 1099 for $100 (but only sent you $65), then you could deduct the $35 as business expenses. But, you would have to report the $100 as income.  Either way you only pay tax on $65.

To say it another way, you have simply agreed to sell your service for $65, rather than $200 or even $100.  Discounts are not deductible, whether they come directly from you or go thru a third party, like Groupon

I didn't pay... they took the money before they paid me.  

Then if you only report what they actually paid you as income you don't have a deduction.  It has already been deducted from your income.  You are reporting less income.  You are only paying tax on your net income.