The majority of adware on the phone is designed to work around any administrative privileges. It does that by either, not needing admin rights to show certain information on the screen or attaching itself to an existing app that has permissions. If it is sophisticated or just merely annoying depends on the app.
My first advice would be to check your currently running processes and see if there is anything suspicious. If there is, attempt to kill the process and look if it might be tied to another app. After killing or uninstalling the apps, restart your phone and check if the problem persists. If it does, try to install a different anti-virus solution or see if now Kaspersky detects the issue.
If the problem persists, you can try tracking the app files or the process files that launch the said adware. I use ES Explorer for file viewing on the phone, but other alternatives should work just as good. (Unfortunately, if your phone is not rooted, there is a low chance that the files will be in a standard .apk format). Most likely the install might be named as a rogue an innocent app, like a game or utility app. See if you don't recognize something and try removing the files manually.
Lastly, if that doesn't help, you could try considering a factory reset, it is quite annoying and unfortunate, but it usually solves the issues.
Good luck! Hope this helps.