The mower starts on throttle/ choke position. But once I pull it down to fast to slow, the engine shuts off. What do you think is the problem?

Hi,
Well, it sounds as if you know what needs to worked on(cleaned.) Problem is that you do not even identify the manufacturer of the engine. Service manuals are available online for many brands. The exception is Briggs.
Peace,
Clarke

Thanks for answering my question. The engine is made by Briggs & Stratton Intek single cylinder.

Hi,
They guard the posting of their service manuals just as John Deere and, to a lesser extent, Stihl do. You will have either a Walbro or a Nikki carb. If you want to post the model and type of the engine, I will give you a link to the parts breakdown.
Peace,
Clarke

It's YTH 20 B 42 Model of Husqvarna.

Hi,
Well, how much info would you like? Engine IPL is at:
http://bsintek.basco.com/BriggsDocumentDisplay/18lmvEXPnfBhU7y.pdf
And the tractor IPL is at:
http://www.husqvarna.com/ddoc/HUSI/HUSI2012_NAen/HUSI2012_NAen_YTH2042_581490127.pdf
And the owners manual is at:
http://www.husqvarna.com/ddoc/HUSO/HUSO2012_NAen/HUSO2012_NAen_YTH2042_532435685R4.pdf
Peace,
Clarke