According to your screenshot, I assumed that you are using Easy tables with access permission. As I known, when you add a new table under Easy tables, it would automatically create the related Node.js back-end, you could go to App Service Editor (Preview) in the Development Tools section of your mobile app as follows:
For Node.js back-end custom authentication, you need to set auth congiguration for your server side, and build your custom login endpoint to validate the client and generate the JWT token for your client. Here is a similar issue, you could refer to here.
Additionally, you could build the C# back-end by yourself and deploy to the mobile app. For custom authentication, you could refer to adrian hall's book about Mobile Apps Custom authentication. For data access, you need to build table controller for each of your SQL table, you could refer to Implementing Table Controllers.
Note:
You must turn on Authentication / Authorization in your App Service. Set the Action to take when request is not authenticated to Allow Request (no action) and do not configure any of the supported authentication providers.
For client app, you could leverage Azure Mobile Client SDK for connecting with your azure mobile app backend. For more details, you could refer to this tutorial about working with managed client for Azure Mobile Apps.
