Personal aspirations that are mistakes (3,5)
I believe the answer is:
own goals
'are mistakes' is the definition.
(I have seen 'Terrible mistakes' mean 'own goals' so perhaps 'mistakes' could also mean 'own goals')
'personal aspirations' is the wordplay.
'personal' becomes 'own' (eg one's own possessions are personal possessions).
'aspirations' becomes 'goals' (I've seen this in another clue).
'own'+'goals'='OWN GOALS'
'that' acts as a link.
(Other definitions for own goals that I've seen before include "Personal ambitions" , "may embarrass members of the team" , "Bad errors" , "Actions that disadvantage the groups carrying them out" , "'Score them against your own team (3,5)'" .)