After start of race, hold-up involved reindeer (7)
I believe the answer is:
rudolph
'reindeer' is the definition.
(name of one of Santa Claus reindeer)
'after start of race hold-up involved' is the wordplay.
'after' means one lot of letters go next to another.
'start of' indicates taking the first letters.
'involved' indicates an anagram (involved can mean confusing or complex).
The first letter of 'race' is 'r'.
'hold'+'up'='holdup'
'holdup' with letters rearranged gives 'udolph'.
'r'+'udolph' is 'RUDOLPH'.
(Other definitions for rudolph that I've seen before include "Red-nosed leader" , "First reindeer" , "one of Santa's team?" , "Haulier" , "Crown prince who died at Mayerling" .)