Damn shower beginning to irritate Gulf national (8)
I believe the answer is:
bahraini
'gulf national' is the definition.
(national of Bahrain, country on the Persian Gulf)
'damn shower beginning to irritate' is the wordplay.
'damn' becomes 'bah' (both are mild curse words).
'shower' becomes 'rain' (rainfall).
'beginning to' indicates taking the first letters.
The initial letter of 'irritate' is 'i'.
'bah'+'rain'+'i'='BAHRAINI'
(Other definitions for bahraini that I've seen before include "from middle eastern state" , "Native of one Gulf state" , "Middle East person" , "Middle-Easterner" .)