Open System Preferences and select iCloud. Do you have a checkmark next to iCloud drive on both computers? If so, this automatically moves your files from the selected folders and puts them in your iCloud account. Also, anything that is synced to the cloud will then copied from the cloud to your other computers.
Solution 1: turn off iCloud drive on your MBA
Solution 2: turn iCloud drive on but then click on the options button and select only specific folders to sync. The first item is Desktop and Documents folders which is the one you've noticed causing trouble. If there are some files you keep in the Documents folder of your desktop that you'd like to sync but others you don't want to sync with the MBA you can create a folder in your user directory on the desktop for storing what you don't want synced.