Unpredictable API requests latency spikes in my ASP.NET Web API published to Azure Web App

There can be several possible reasons:

The problem could be in your application code, create a staging environment and re-run your test with profiler tool telemetry (i.e. using YourKit .NET Profiler) - this will allow you to detect the heaviest methods, largest objects, slowest DB queries, etc.Also do a load test on your API with JMeter.

I would recommend you to try Kudu Process API to look at the list of currently running processes, and get more info about them list their CPU time.

The article for how to monitor CPU using in Azure App service are shown below:

https://azure.microsoft.com/en-in/documentation/articles/web-sites-monitor/

https://azure.microsoft.com/en-in/documentation/articles/app-insights-web-monitor-performance/