I think I will disagree
Oil is a big factor and obviously using the right oil is important although I believe in your case its not so much the grade as the Punto Sporting and HGT where designed for 10W 40 and all tollerences within the engine would of been set for that grade when manufactoring.
In my opinion whats needed is regular oil changes around 6 months just to keep things nice and clean!
As for the tappets, I believe the 1.2 16V has hydrolic tappets as found in the 1.8 16V HGT?? So the tiny oil channels may be clogged, It happens
Get some new oil such as Shell Helix Plus, new Oil filter and some Wynns Hydraulic Valve Lifter (Professional type) , empty old oil down to just above minimum on the dip stick and enter the Wynns Hydraulic Valve Lifter into the oil directly.
Let engine tick over until hot and then rev the engine to around 3000 rpm for a few mins and continue this for a few days or alternativly add a little more oil and drive around with the cleaner in...

You should notice silent tappets within 15 - 30 mins of first putting in and running the car anyhow!
Perform the new oil change after a few days of the Wynns Hydraulic Valve Lifter being in and you shouldnt have anymore issues as long as you stick to regular oil changes! Please note if your engine is really dirty inside I wouldnt advise this as alot of gunge may break free and clog your engine more
Cheers