Unable to add deleted songs downloaded from a CD back to iPhone music

I downloaded music from CD's to my MacBook in the MUSIC library. Then, I added those songs to my iPhone in Music library. I accidently deleted some songs from my iPhone. When I connected my phone to my MacBook to add those songs back to my music library, they are not able to sync back into the Library and don't show up in the library.

I now see that when you "delete" songs from your music library on your iPhone, those songs are not really deleted. When I connected my phone to my computer and go to the Music ap, I click on the iPhone under devices for a list of the songs on my iPhone. Those songs that I deleted show up in the list of songs on my iPhone. Yet, I am unable to play or see them in my music library on the phone.

Is there a fix for this? Or, did Apple do this on purpose to make people buy music from iTunes. If so, that's dirty.