Music without subscription

On an iPhone, iPad, or iPod touch

1. Open the iTunes Store app.
2. On your iPhone or iPod touch, tap  More in the menu at the bottom of the screen, then tap  Purchased. On an iPad, just tap  Purchased. 
3. Depending on what you want to redownload, tap Music, Movies, or TV Shows. If you use Family Sharing, tap My Purchases or choose a family member’s name to see content that they purchased.
4. Find the movie, TV show, or music that you want to redownload, then tap it. 
5. Tap  to the right of the title. The movie, TV show, or music downloads to your device. You can find movies and TV shows that you redownload in the TV app§ or Videos app, and music in the Music app. Don't see the item that you purchased?

Music from the Apple Music subscription is only available to you while you are subscribed.

Music from iTunes Store purchases are unrelated to subscriptions, and require a one time payment. These purchases are normally downloaded and added to your library. You may re-download previous purchases linked to the matching Apple ID from the iTunes Store (if these are still for sale there).

Music from other sources (e.g. converted from audio CDs) have to be synced using iTunes on a computer. A new sync may remove tracks that are not in the selection.

There are two separate music libraries. The iTunes Store library has songs you purchase (actually pay for) and download: you can keep them permanently.

The Apple Music library (which is slightly smaller) has songs you can stream or download (but not move or copy) as long as you have a subscription (first three months free). If you stop subscribing you will no longer be able to play the music. If you don't cancel the free trial you will automatically be charged at the end of it: your access to the songs will disappear as soon as you cancel it. If you start paying the subscription and later cancel it you will have access until the end of the current subscription month.

It rather sounds as if you had the free trial, and downloaded some songs, but then cancelled it in order not to be charged for the subscription. You can still purchase songs on an individual basis (to keep permanently) from the iTunes Store: if you want to resume access to Apple Music you can subscribe (but you won't get another free trial, of course).

I think you're at the point where you should contact iTunes Support: iTunes Store - Contact Us - Apple Support

That's it: it's really two separate services.