2017 Tohatsu MFS6C coughs out carburetor intake when restarting

2017 6HP MFS6C 4-stroke
In general, the engine is running very well.
Idles nicely, responds to open throttle, cooling water flow good, shifts crisply.
New spark plug, engine oil change, gear oil change, new impeller.

After warmup, with the twist grip set to the RESTART position and choke knob in,
it sometimes "coughs" and doesn't "catch". There is some smoke that comes out
of the carburetor intake screen. The whole engine usually kicks slightly to one side.

Shall I bump the idle speed up? I don't really know what's correct as I don't have a tachometer.
Anything else I should try?

I'm concerned that if this continues, it will damage the piston or connecting rod......

I noticed something else...from a cold start, after several minutes, the idle speed goes up significantly (all of a sudden).
I'm wondering if the thermostat is bad. Perhaps the engine is never warming up and that is what is causing the coughing
and difficulty restarting. Sound plausible?

Well, I figured it out. I wasn't letting the engine warm up enough before trying to RESTART.
Idling for 3~4 minutes isn't enough. It has to idle for 12~15 minutes before it is warm.
Then, the restart procedure will work. I guess the long warm up time is because it is
a 4-stroke engine...uses very little fuel at idle! Not sure why the idle speed was jumping.
Perhaps the idle was set too low. Doesn't seem to be doing that at the present time.