I too use an old Honda HR-214. I have been using it for about the past 8 years, but I suspect that it is at least 15 to 20 years old. It has always smoked at startup.
This Spring, I used 10W-30 Mobil-1 when I changed the oil. At that time, I also changed the spark plug and cleaned the air filter. It ran as well as it always had, but it still smoked at startup.
Earlier this Summer, I read Bill Kapaun's note about loosening the oil cap after shutdown. That seemed to help some. I just loosened it and re-tightened it. I didn't leave it off for half an hour as his post in this thread recommends.
In the middle of August, I had some Redline SI-1 fuel system cleaner leftover that I added to the two gallon gas can I use for the mower. I was a fairly strong dose, I think two to three times their recommendation for a maintenance dose. The smoking got worse for a time or two, but in the past two times I started it, it seemed to be getting better. The last time I started it, I had not started the mower for two weeks, and usually it is worse after a long rest. I have only used about half of the gas in the can, but I should have mostly finished it by the end of the season.
I guess the real test to see if this fixed the problem is how it runs next year.
Good luck to all.