Rather wet philosopher failing to finish papers (5)
I believe the answer is:
humid
'rather wet' is the definition.
(I've seen this before)
'philosopher failing to finish papers' is the wordplay.
'philosopher' becomes 'hume' (David Hume).
'failing to finish' means to remove the last letter.
'papers' becomes 'ID' (identifying documents are one's 'papers').
'hume' with its last letter taken away is 'hum'.
'hum'+'id'='HUMID'
(Other definitions for humid that I've seen before include "Moist (atmosphere)" , "Damp, steamy" , "Clammy" , "Warm and steamy" , "Describes air that contains a great deal of water vapour" .)