I just want to backup my notes

You can sync Notes, and only Notes, between devices in iCloud by going into Settings > [your name] > iCloud and toggling off everything except Notes. iCloud does not back up Notes.

Open Notes on the device (iPhone) you deleted the note with, then click or swipe all the way back until you can't anymore. You should see a screen titles "Folders". From here, click "Recently Deleted". It has a trash can icon on it. If the note appears here, you can restore it by clicking on it, then clicking the folder icon at the bottom. If the note doesn't appear, I'm afraid it's gone.

If you deleted the Notes from your device, check the Recently Deleted folder. They may appear there. If not, they've been permanently deleted.