Girl falling just short of the record? (4)
I believe the answer is:
anna
'girl' is the definition.
'falling just short of the record?' is the wordplay.
'falling just short of' means to remove the last letter.
'the record?' becomes 'annal' (I've seen this before).
'annal' with its final letter taken off is 'ANNA'.
(Other definitions for anna that I've seen before include "Book by Tolstoy" , "copper, once, in India" , "Former Indian coin; girl's name" , "Former one sixteenth of Rupee" , "worth very little in India" .)