If I have self-employment expenses that I did not claim on 2021 taxes, can I claim them now?

No.   You cannot use 2021 business expenses on a 2022 tax return.  You can amend your 2021 tax return to include them.

See this TurboTax support FAQ for amending a tax return - https://ttlc.intuit.com/turbotax-support/en-us/help-article/tax-return/amend-change-correct-return-a...

No, as xmasbaby0 stated, you can not include 2021 business expenses on your 2022 return.  The 2021 expenses would need to be added to your 2021 return. To determine whether you can deduct an expense, ask yourself: Is this expense both ordinary and necessary to the business? The IRS requires both elements.

1. An expense is ordinary if it is common and accepted in your industry.
2. An expense is necessary if it is helpful and appropriate for your business. 

If you need to amend, you will need to follow the following steps:

1. Sign in to TurboTax
2. On the home screen, scroll down to Your tax returns & documents and select the year you want to amend
3. Select Amend (change) return, then Amend using TurboTax Online
   • If you're told to amend your return via TurboTax Desktop, follow the steps for TurboTax CD/Download below instead
4. On the screen OK, let's get a kickstart on your amended return, select the reason(s) you're amending and Continue
5. When you reach Here's the info for your amended federal return, select Start next to the info you need to change
6. Continue through the screens, make the changes you need to make, and carefully answer the remaining questions to finish amending your return

For more information, see the link below:

How do I amend my federal tax return?

Taking Business Tax Deductions

@hbott