I have recently successfully cleaned my hard drive and restored data with the TimeMachine backup. Now I would like to prepare backup of "new" system. Do I have to clean the drive first?
If you're asking about multiple backup of the same Mac on another backup volume, then no, that's not possible as the Time Machine will backup the same Mac and effectively overwrite the most recent backup.
If on the other hand you're asking about backing up multiple (different) Macs on the same backup volume, that's possible. I back up 5 Macs onto my NAS regularly.
No, you don't have to clean it. Just plug it in and a new Backup will be written. As far as I know the most recent backup is used to restore the system, so if you consider to restore the very same state again in the future, the only solution is not to plug in the time machine hard disk after you backed up your current state.
However, you should consider that this is not how time machine is intended to be used.