Clue: High card following lower one ? It may facilitate finesse (6)
Answer: TENACE
Explanation: Definition: It may facilitate finesse
**ACE** ("high card") *following* **TEN** ("lower [card]") 
 
 
*In bridge, a finesse is an attempt to win a hand with a lower card than another in one's hand, and menace is a combination of cards that allows this move to take place. I don't play bridge, so that may or may not be an adequate explanation – apologies if not!