"Cut" isn't a filesystem operation, so I assume that your file manager interprets "cut and paste" as "move". In which case there is only one difference between "move" and "copy then delete" and that is that if something goes wrong with a copy you still have the original until you delete it, while if something goes wrong with a move that's game over. But if you do both without checking the copy there is no difference.
As for file recovery, it's not generally very successful on Android (and of course if the phone has been used normally in the meanwhile there is a risk the file has been overwritten anyway). The one thing you have going for you is that this one won't be in an area that needs root to access it (which would be the case for deleted messages) and you've not reset the phone (in which case encryption keys are gone as well). So it's possible that an Android file recovery app might be able to recover it (standard file recovery software over USB won't work because the USB connection is MTP rather than USB mass storage). Look in the Play Store for something that doesn't require root and give it a go - there's a chance it will work, and as long as the app itself isn't adware not much to lose.
Of course if the file was stored on a microSD card then it's simple: remove the card from the phone, stick it in a card reader (they are cheap) and run whatever file recovery software you like.
