Po code 340 for bad camshaft synchronizer been driving on it for 3 weeeks car finally broke down but their was also smoke under hood what that have anything to do with that or something totally different

The smoke comng from under the hood may not have anything directly related to your cam code. Could be a valve cover gasket but its hard to say without looking at it. Here is a link to find a shop near you. https://repairpal.com/repair-shops Alex