Azure App Service is a service used to create and deploy scalable, mission-critical web apps.
yep - you can safely delete the awverify CNAME record from your DNS settings after Azure successfully validates your custom domain.
The awverify record is only used by Azure during the initial configuration and ownership verification stage. Once Azure shows the custom domain as verified and active, the validation record is no longer checked or required by the platform.
Just for the clarity sake (I gather you already know this) make sure not to delete the actual production CNAME or A record, such as one pointing your www or root domain to your azurewebsites.net app. That record is required to route live visitor traffic.
For confirmation (from Sam Cogan), refer to https://serverfault.com/questions/783103/remove-cname-after-azure-has-verified-the-domain
If the above response helps answer your question, remember to "Accept Answer" so that others in the community facing similar issues can easily find the solution. Your contribution is highly appreciated.
hth
Marcin