How to Stop Recently added songs from appearing in iPhone Music app

OK.

What is it you’re calling the “ Main screen”?

This below in my screenshot in the black font? “Recently Added”. I swiped up a bit to show you, otherwise mine is “hidden”. You can’t get rid of that. It’s just there. 

It’s not really on display, you have to swipe up to see it. Well with mine it is because I have a longer list of which lists I want to see.  It’s the only way to sort of hide the recently added album covers - Add more things to your red font.You can only edit what’s being displayed in the red font, by using the edit button - top right.

I do see once you tap in downloaded Music, it’s there again, but once again you can edit THAT list as well to add more things to push it down from view. 

if it’s bothersome to you can try telling Apple but unless ,there’s thousands of others in agreement, it likely won’t get changed. Send Feedback: https://www.apple.com/feedback/apple-music.html. Now that you mention it I don’t like it either, otherwise I wouldn’t have set up my screenshot to hide what’s there!

It might be best to just make your own playlists from the downloaded. This will assure you get the songs of your choice. The others won’t be in there at all. That’s what I do. The hands- free method of adding to playlist is to just ask Siri “add this song to my xxxx playlist”

it’s a workaround to your dilemma, so give it a shot.

You can choose whatever you like to be shown in your Library by editing the library view.

Go to the Library Tab, on the top right you will see Edit - tap that and check what you want shown. If you only want to see downloaded music , just select that. Then when you want to use it, start from that list.

Additionally you can also edit that list as well.

If that’s what you mean?

Or

Add a song to a playlist in Apple Music without adding it to your library

https://support.apple.com/en-ca/HT206183

How to Use the Music App

https://support.apple.com/en-us/HT204951

You can also Send Feedback 

https://www.apple.com/feedback/apple-music.html