I have a Flymo petrol mower. Once I have started the engine running, it will run a short while then will die off. It will start coughing and engine won't accelerate.
I also noticed that it gives out black smoke every now and then coming out of the exhaust.
Your engine is running too rich after it warms up.
Check the air filter to see if it needs cleaned or replaced.
If the filter is OK the carburator may need adjusting and/or cleaning.