Where cattle may be kept near river Exe at first (4)
I believe the answer is:
byre
'where cattle may be kept' is the definition.
(byre means a building for cows)
'near river exe at first' is the wordplay.
'near' becomes 'by' (I've seen this in other clues).
'at first' says to take the initial letters.
The initial letters of 'river exe' is 're'.
'by'+'re'='BYRE'
(Other definitions for byre that I've seen before include "farm building" , "Cow barn" , "lower in here" , "Cattle-shed" , "Shelter for cows" .)