Our newly bought refurbished Sage Bambino Plus gave us the single and double shot buttons (1 cup and 2 cups) intermittently flashing, indicated that the machine thinks it needs backflushing. We have the newer backflush disk without the hole of the earlier disks. I put a tablet on it and pressed and held the single and double shot buttons at the same time for at more than 5 seconds till the two shot button lights both became solid. I started the backflushing with the single shot button and it began its cycle. It ended realtively fast after only a few shots and gave me all buttons lighted as the sign that the cycle is over. But the tablet was practically still intact and the water in the tray was as clear as before.

But after turning the machine off and later on again, the two shot buttons were intermittently flashing again. Just for the sake of it I rerun the backflushing ending with the result that the machine is ready again. But nothing has changed. The single and double shot buttons are once again intermittently flashing. But to my surprise the machine is still working: I only need to press the steam button then all the 5 lights go on. Now I can pull a shot.

Do I need to reset the machine to get rid of this strange behaviour?

Some instruction are wrong in the manual so it is important to follow the video instruction on Breville website instead of the user manual (it also has some typos). You can also write Breville Support and they will send you the latest instructions.
For the cleaning tablet, it is normal. Personally I prefer to use powder cleaner instead of the tablet.