Last part of race perhaps flat and neat (4,8)
I believe the answer is:
home straight
'last part of race' is the definition.
(home straight means the final stretch of a race)
'perhaps flat and neat' is the wordplay.
'perhaps flat' becomes 'home' (a flat or apartment is a kind of home).
'and' says to put letters next to each other.
'neat' becomes 'straight' (synonyms).
'home'+'straight'='HOME STRAIGHT'
(Other definitions for home straight that I've seen before include "Final stage of a racecourse" , "covered in neat report" , "final stage of race" , "Last stage of race" , "Last part of a race" .)