I recall that Gulftream was involved some years ago in a hushkit proposal (must have been Spey) that involved a large aft duct shrouding a free turbine connected directly to an outer concentric "bypass" fan. The resultant assembly was expected to meet Stage III noise while reducing SFC and increasing thrust. Don't believe it ever reached the prototype stage.
Maybe this is helpful in understanding the answer to the question that runs through this thread. Would the shroud/turbine/fan assembly itself have generated thrust? No, the engine generates thrust, but the hush-kit assembly captures and organises energy from the engine exhaust that would otherwise be lost, and thereby converts it into thrust which is transmitted to the engine through the husk-kit attachments.
Surely it is the same in principle with a winglet?