The usual wash-up for the perplexed and mystified:
ACROSS
1A. last letter from "skirt" = T + raised = HIGH. 4A. anag. DEN. 6A. insertion of one = I into diplomatic relations = TACT. 7A. delete half of: showing strength and determination = RUGGED. 8A. reversal of lip = RIM + first two letters of "the".
DOWN
1D. finger (e.g. of whisky, etc) = TOT, + reversal of tuber = ME. 2D. a dog's dinner may very well end up IN CUR. 3D. double def. 4D. first letters from "explore and rediscover". 5D. partial, from finger = DIGIT.