Is there an iPhone app (as opposed to desktop software) that shows actual battery cycles - cycles number
No. For a very long time, the API for that information is private. Any apps on the App Store that manage to trick review or access private API generally get shut down by Apple review / oversight in days or weeks. 

The official way to get that data is through Apple Support. 


https://support.apple.com 
https://www.apple.com/batteries/service-and-recycling/


Until and unless Apple decides to provide cycle count detail in the settings app for iOS or in a public API, you will likely need a trusted computer and / or jailbreak to get those details. 


How do I check an iOS device&#39;s battery health?