Synthetic oil will not make your mower smoke. Did you add it to what was already in there, or completely replace the oil?
There is no point in adding synthetic oil to existing mineral oil, as the presence of mineral oil will negate any advantage.
The most likely explanation is that there is now too much oil in the crankcase. Check and adjust. Has the machine been tipped? If so, just allow it to run till the smoke clears (possibly 10 mins or more)
Hope this helps - if not, please come back here and advise.
What make of engine is fitted? I was assuming a Briggs & Stratton, but some makes are more sensitive to the wrong oil than others. How old? The correct oil for all 4-stroke mower engines is SAE30. Synthetic is usually 5W50 or similar. This is much thinner than SAE30, and if the engine is more than very slightly worn, it will smoke due to oil leaking past the valve guides.
Replace the Mobil-1 with standard SAE30 mineral oil. If you still have a problem, let me know.