Play Apple Music without library on external hard drive connected

Is there a way to play my music (I am subscribed to Apple Music and iTunes Match) if my library and music, which I keep on an external hard drive, are not connected?

I have recently got a MacBook Air but the storage on it is only 128GB. I have therefore transferred all of my iTunes folder (which includes my Library, Media Folder etc) from my old Windows laptop on to an external hard drive. I have connected this to my MacBook and mapped iTunes to it and all is fine when the External Hard drive is connected. However, on occasion I'd like to be able to play music on my MacBook without having to have the hard drive attached. I perhaps naively thought that if I had synced my library to iCloud and have Match turned on, I could do this...? So far when I have tried, my songs (apx 18,500) appear but when I try and play one of them I get an error along the lines of "...could not be used because the original file could not be found. Would you like to locate it?". If I cancel that message it seems to then play it via Apple Music which I guess is fine for one song at a time, but not if then want to play an album/playlist etc. Am I able to have all my songs available via the cloud?

If not possible as described above, is there another way I'm able to do it...such as having a local library? I want to avoid having to store the media files on my MacBook hard drive.

Thanks