If Tarkin blanks this objective, do the captured cards get discarded? 

We don't ask such questions. But I would guess not seeing as how Dirty Secrets doesn't remove captured cards from scum objectives.

No, why would they? The effect resolves, places cards as captured here and that's the end of it. LS missions are DS objectives and game rules allow DS objectives to have cards captured at them.

I didn't know if captured/facedown cards count as enhancements or not. 

They don't. Captured cards are one thing, enhancements are another, and generic facedown cards are another. And game rules allow objectives to have captured cards and enhancements, so blanking an objective doesn't get rid of either (unless, for example, an enhancement said "Enhance an elite objective" - blanking the objective would remove elite and thus it would no longer be a legal target for the enhancement). Facedown cards are a different thing in that the rules don't mention them and it seems like the game doesn't check whether it's legal for a card to have facedown cards attached, so Tarkin blanking such a card doesn't discard the facedowns. In the case of Hero's Beginning and Imperial Bureaucracy Tarkin does absolutely nothing because the ability to use the facedown cards is a lasting effect set up when the Reaction initially resolved, and Tarkin doesn't touch already initiated lasting effects. In case of Hidden Grove and R2-D2 a Tarkin blank removes both the ability to add new facedowns and the ability to use existing facedowns (because there the ability to use them is a constant effect, not a lasting one), but the cards stay where they are.