Thanks for asking question! When you enable mutual auth for your application, all paths under the root of your app require a client certificate for access. To remove this requirement for certain paths, define exclusion paths as part of your application configuration.
Exclude paths from requiring authentication
Also, You want to know in App Service, TLS termination of the request happens at the frontend load balancer. When forwarding the request to your app code with client certificates enabled, App Service injects an X-ARR-ClientCert request header with the client certificate. App Service does not do anything with this client certificate other than forwarding it to your app and your app code is responsible for validating the client certificate.
Please refer to this document for Configuring TLS mutual authentication for Azure App Service might be helpful:
https://learn.microsoft.com/en-us/azure/app-service/app-service-web-configure-tls-mutual-auth
Let us know incase issue persists.