Primary heat, fuel, and smoke flavor source in a stick burner is wood split like fireplace logs. Usually ignited with propane torch and sometimes hot charcoal then kept hot during the smoke by adding additional unburned wood logs.
Primary heat and fuel source in charcoal smokers is pre-burned wood like lump charcoal or pressed briquettes. Very little smoke flavor comes from the charcoal because the charcoal making process carbonizies the wood, burning off all the flavor characteristics in the wood. Consequently unburned wood chunks the size of golf balls up to fist sized are used to add specific wood smoke flavors to the meat.
Assuming you have a charcoal smoker with a side firebox, yes, adding unburned wood to your hot fire will cause white or grey smoke, but it usually doesn't last long on a hot fire. I use larger pieces when I initially fire up the my charcoal smoker, then smaller pieces later in the smoke as needed. White smoke doesn't last as long with the smaller pieces later in the smoke.