Doctor arguing with A&E drops monitor (4,5)
I believe the answer is:
rain gauge
'drops monitor' is the definition.
(a rain gauge monitors drops of rain)
'doctor arguing with ae' is the wordplay.
'doctor' indicates anagramming the letters.
'with' is a charade indicator (letters next to each other).
'arguing'+'ae'='arguingae'
'arguingae' with letters rearranged gives 'RAIN-GAUGE'.
(Other definitions for rain gauge that I've seen before include "The measure of a meteorologist" , "Pluviometer" , "one can measure water" , "met recorder?" , "Meteorological instrument" .)