Admit ambition is an embarrassing mistake (3,4)
I believe the answer is:
own goal
'an embarrassing mistake' is the definition.
(I know that own goals are often embarrassing)
'admit ambition' is the wordplay.
'admit' becomes 'own' (own can mean to admit or acknowledge).
'ambition' becomes 'goal' (I've seen this before).
'own'+'goal'='OWN GOAL'
'is' is the link.
(Other definitions for own goal that I've seen before include ""Wrong-net" score" , "that the opposition may thank you for?" , "Defender's faux pas" , "embarrassing error" , "move that backfired?" .)