a backfire through the carb can only be caused by a serious misadjustment in timing. it would take the timing being out by about 180 degrees for this to happen.
a "cough" out of the carb is a classic symptom of a lean air/fuel mixture. a lean mis-fire.
if this is the classic lean mis-fire, as I suspect, and the jets have not been changed since before this condition appeared, the usual cause is a vacuum leak at the intake manifold seals, either where the carb attaches, or where the intake manifold attaches to the head. another possibility is a vacuum diaphram leak in the carb.
Dealer called Suzuki japan, they checked your suggestions, They truly do not know what is causing it. it is more of mis fire ( cough) the dealer called it a backfire through the carb. This machine ran great until the 100 hour tune up recomended by Suzuki. I just don't know neither does the Dealer service dept. any other suggestions.
PS. It could be 80 degrees and I could be running for 1/2 hour and it usually does it when you first start or sometimes when your trying to run at a steady speed
you have a lean air/fuel mixture.
you either have vacuum leak, low compression or a restricted fuel supply resulting in a low float level.
check the compression and check the fuel delivery.
all I can do is tell you what causes a lean misfire.
I can't tell you what happened at the dealership....I wasn't there.
if you want to fix your problem you are going to have to help me help you.
if you are unwilling to do that, let me know and I will opt out now.
black plugs and lean misfire indicate low compression.
let me know if you need more help.