I can use Coconut Battery on my Macbook which shows cycles number on my iPhone when I plug iPhone into Macbook. I don't want to use that however. I want to have an iPhone app for that. Do you know any?
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.
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.