Miss first half of game for children's activity (8)
I believe the answer is:
skipping
'children's activity' is the definition.
I can't tell whether this defines the answer.
'miss first half of game for' is the wordplay.
'miss' becomes 'skipover' (skipping over is a kind of missing).
'first half of' suggests halving the letters.
'game for' becomes 'ping' (I am not sure about this - if you are sure you should believe this answer much more).
'skipover' cut in half is 'skip'.
'skip'+'ping'='SKIPPING'
Can you help me to learn more?
(Other definitions for skipping that I've seen before include "Playing with a rope" , "Cutting" , "Missing - physical exercise" , "Jumping lightly or bypassing" , "amusement" .)