Fear losing face, thus making a mistake (5)
I believe the answer is:
error
'a mistake' is the definition.
(I know that mistake can be written as error)
'fear losing face' is the wordplay.
'fear' becomes 'terror' (terror is a kind of fear).
'losing face' means to remove the first letter.
'terror' with its first letter taken away is 'ERROR'.
'thus making' acts as a link.
(Other definitions for error that I've seen before include "Mistake from terrorist" , "A mistake or inaccuracy" , "Howler" , "1 down" , "Bug" .)