A few thoughts that may provoke debate . I am not an expert. They may not be relevant. I may be wrong on any of them .
I believe Honda takes measures to ensure faster warm up such as changing timing and it could include not charging when the car is stationary .This may be more as a fuel saving measure . Extra load would tend to produce more heat.
At 2-3c the effect of air passing through the radiator will be considerably colder due to wind
chill. . The engine ,radiator coolant etc may not be fully warmed up in 3 km or so.
Modern cars dont just measure the engine temperature when metering the ideal fuel/air ratio. They also monitor the ambient air temperature and rate of flow of air into the engine. Because colder air is denser and more concentrated. So the car 'knows' its cold and may continue with its special measures longer than you expect even when the engine may be warm.
Because the engine often runs relatively slowly , or not at all , compared to the speed of the car significant wind chill might cause the radiator coolant to become locally very cold . This could lead to problems when the engine restarts. To minimise this its possible the car may use an electric coolant pump , or an engine driven one that couples or uncouples thermally or electrically. I cant find any technical information on whether this is the case, but it makes sense. It could keep warm coolant circulating, or stop it circulating when thats better. And reduce load on the engine when its not needed . (If you havnt got one Honda you heard it from me first. I want paying
)
In warmer weather thermosyphon circulation may sometimes be sufficient .
A bit like radiator fans evolving from constant, to thermo coupled to electric.
If such a system is used this may be the noise you are hearing. Either the pump running or a coupling freewheeling or something. A further thought here. Air con compressors uncouple in this way and sometimes rattle. Could a change in temperature/climate control settings or activity be the cause?
Petrol is less volatile in colder weather. This could lead to rougher running. And any ethanol makes this worse. E10 is probably not a problem but I know that in cold climates E85 (85% ethanol) is changed to something like 70% ethanol in winter - but still called E85
I think the formula of petrol is changed in winter so its more volatile and vapourises better.(This is definitely true with diesel, but also true with petrol I think, at least in some countries. )
Cars running poorly when cold at this time of year might be because you ,or the filling station ,still have summer grade petrol . Things may improve next fill.