Air's foul with daughter rarely having a shower (4)

I believe the answer is:

arid

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'rarely having a shower' is the definition.
I can't judge whether this definition defines the answer.

'air's foul with daughter' is the wordplay.
'foul' is an anagram indicator.
'with' means one lot of letters go next to another.
'daughter' becomes 'd' (genealogical abbreviation).
'air' anagrammed gives 'ari'.
'ari'+'d'='ARID'

Can you help me to learn more?

(Other definitions for arid that I've seen before include "Dry and sere" , "Parched with heat" , "Dry raid" , "Dry and barren" , "Drought-hit" .)

I've seen this clue in The Independent.

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