Specialist almost spoilt niece's encounter (10)
I believe the answer is:
experience
'encounter' is the definition.
(I've seen this before)
'specialist almost spoilt niece's' is the wordplay.
'specialist' becomes 'expert' (I've seen this before).
'almost' means to remove the last letter (most of the word but not all of it).
'spoilt' indicates anagramming the letters.
'expert' with its last letter taken away is 'exper'.
'niece' with letters rearranged gives 'ience'.
'exper'+'ience'='EXPERIENCE'
(Other definitions for experience that I've seen before include "Participate in - knowledge and understanding gained" , "Live through; feel" , "Feel (an emotion)" , "Go through" , "Undergo" .)