That is a bad idea. Time Machine is strictly a backup utility. What you describe will require archiving files, which Time Machine is not designed to accomplish.
Once you erase files from your MacBook Air, they will eventually be erased from your Time Machine backup. This is by design, as Time Machine will eventually fill your backup volume and will delete old files as required. Depending on when they were first backed up and how long they existed on your MBA before you erase them, they could disappear in as little as one hour.
Use Carbon Copy Cloner or equivalent "clone" utility and configure it to archive files that do not exist on the source, or simply copy them over manually using the Finder. You can then delete them from your MBA. Bear in mind that once you do that, you no longer have a backup copy. The only copies will exist on the external HD.