Part of racecourse tackled with apparent solemnity (8-5)
I believe the answer is:
straight-faced
'with apparent solemnity' is the definition.
(looking serious or solemn)
'part of racecourse tackled' is the wordplay.
'part of racecourse' becomes 'straight' (as in the final straight).
'tackled' becomes 'faced' (face can mean to tackle or deal with).
'straight'+'faced'='STRAIGHT-FACED'
(Other definitions for straight-faced that I've seen before include "Without smiling" , "With an expression to conceal amusement" , "Resisting the temptation to laugh" .)