Clean the Inbox means: remove all the stuff which is currently there. Delete it or move it anywhere else (this means: into another folder) but don’t let this there in the Inbox. The Inbox is not at all a good place for archiving read mails. Make an Archive folder for that on the mail server, I have some ten sub folders in my Archive. The best way to do this is using the web interface of your mail server on a real computer. Some mail apps store mail messages including attachements locally, these stored data get sometimes in disorder when you move too much around.
Push mode is a feature of the mail account(s). You must (in the mail app) open the settings of a mail account. There is a section for settings of receiving. When a mail account is in Push mode the frequency of receiving new messages from the server is typically set to “Never”. This is because the mail app receives new messages by a push from the server. Otherwise if you see there an interval of 5 minutes or so your mail app requests all these data from the mail server every five minutes. This is called polling.
The Push mode causes much less data transfer, 100kB per day or so. If your mail server(s) support(s) it you should setup each mail account for Push. It’s also much faster.
(Note: setting up a mail account for Push needs some more options than the polling frequency mentioned above.)