I would suggest check the valves, when cold, and make sure they are right. As for the sputtering and backfiring above idle, did you go through the timing advancer? The springs on it get weak, which causes the timing to not advance on acceleration. I had this problem on my 200ES, very similar symptoms, running rich and pops and sputters above idle. I went through the carb and such and put a timing light on it and found out the timing was stuck in one spot, i had a new set of springs for the advancer stashed away and put them on and that fixed it up. Also take the advancer out, take it apart, clean it, oil all the moving parts, make sure the weights move freely, etc.
If neither of those help... then i am in the same place you are scratching my head.
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1984 Honda ATC200ES "Big Red"
1982 ATC200E "Hondie"
1988 TRX300FW "Project Quad" Still in progress....