Modern Vespa

I'm having an issue with my MP3 cutting out when the throttle is closed and the bike comes to a standstill. It is intermittent and is far worse just after the bike has been started and the engine is cold. The issue also seems to disappear when the bike undergoes periods of heavy use....(such periods of usage are infrequent...).

I can see that this isn't an isolated issue with the MP3 so before I dive in, I wanted to make use of the forum's knowledge...

In terms of air...
I would like to clean the throttle body- is it necessary to remove the plastics/underseat to do this? It looks a bit tricky otherwise...

The air filter isn't old but I will change it anyway.

Could there be an issue with the lambda sensor? Is it worth replacing the silencer gasket/looking for an issue there? The silencer doesn't have any holes in it although changing it wouldn't do any harm as it's seen better days..

As for the fuel system...
The injector was replaced about 5k miles ago. Is it worth cleaning that and the fuel system with some redex type solution?

Is there anything else that I should consider with the fuel system?

It's not the worst problem in the world but I could do without the bike cutting out and leaving me in a dangerous situation. It can definitely get the old adrenaline pumping at times....

Any advice/information on what might help me to resolve the matter would be most appreciated...

Sounds like the bike is (over)due for its valve adjustment.

Your symptoms fit perfectly for a engine with tight valve(s)

Clean the idle port on the throttle body with Throttle Body Cleaner. Just pull off the air cleaner and the inlet hose to the throttle body. Then look for the small port on the left side. Rev the motor and spray some throttle body cleaner into that hole a few times. If motor stalls just start it and keep cleaning.
Good Idea to do an oil change after this too to remove excess cleaner that gets past the rings.

What year model is it, the some early models hade fuel pump issues, and many miles on it? Clean the cvt system if there is lot of dust on the variator slide tube it wil move slower when letting throttle off,

Also TPS misreading might cause this.