This may be the cause of difficult cold start - engine is not getting more air when it is cold, so it is getting more fuel (CSV and solenoid work), but not more air as the air passage is clogged - too rich to run.
You got to WRD device. Some of these parts are hard to get. You may want to attempt cleaning. Just do not clean the air valve piston and its bore with sand paper, they need to be a tight fit to be able to cut off air.
Please look through the Technical Manual and posts on WRD:
https://www.sl113.org/wiki/Engine/WarmRunningDeviceIt is not an air pump. In essence, when engine is cold, the air valve inside is open and allows lots of air to be sucked to the engine. The air goes through a filter that is inside this "gold" tin. There is a paper insert there - you may want to buy this filter and replace it as well. As engine warms up, thermostat expands, air valve piston moves and blocks the air passage - when engine does not need so much air any more. That is why the piston need to fit its bore - if it does not, air goes through even when engine is warm, causing the lean condition. It is easy to check if air is coming through or not - unscrew the filter and feel with your finger if there is suction there.
But judging from your picture the air passage is completely clogged. I would clean it, not destroying the bore walls surface nor the piston surface - they need to be a tight fit.
When putting it back together, it is good to put some sealant between the thermostat and its housing to prevent coolant from leaking down - as might have happened in your case.
If you manage to clean it, replace the filter, check and replace the thermostat if need be (check: should be expanding ca 5mm when you put it in a kettle with hot water), put it back together with some sealant as I wrote - you should be ok.