Any caulking or adhesives better than others?

So we had a massive leak through our front porch. Had it repaired but the contractor never put in a drip edge. So now I'm finding small water marks below it. I'm wondering if there's a caulk I can use to put a small bead to act as a drip edge until I can get the time to remove the layer of stone and cut in a drip edge myself next summer.

Thanks