We are married filing jointly. I have income from self-employment. I paid my wife for some work. Can I add this payment as earned income for my wife and expenses to me?

Yes.  As long as you would have pad a "stranger" to perform the same work that your wife did.  You will report this expense as "Contract labor" on your Schedule C, Line 11.

"Enter the total cost of contract labor for the tax year. Contract labor includes payments to persons you do not treat as employees (for example, independent contractors) for services performed for your trade or business. Do not include contract labor deducted elsewhere on your return, such as contract labor includible on line 17, 21, 26, or 37. Also, do not include salaries and wages paid to your employees; instead, see Line 26, later. 

You must file Form 1099-MISC to report contract labor payments of $600 or more during the year. See the Instructions for Form 1099-MISC for details."

You will report your wife's income as self-employed income.

To report self-employed income without a 1099-MISC:

1. Open your return in TurboTax.
2. Click on “Take me to my return” if applicable.
3. Search for "self employed income" in Search box.
4. Click on "Jump to self employed income" link.
5. Answer/follow the screen interview prompts.