Weariness in last bit of The News Quiz essentially (5)
I believe the answer is:
ennui
'weariness' is the definition.
(ennui is weariness or boredom)
'last bit of the news quiz essentially' is the wordplay.
'last bit of' says to take the final letters.
'news' becomes 'nn' (an abbreviation for new written twice).
'essentially' means to look at the middle letters.
The centre of 'quiz' is 'ui'.
The final letter of 'the' is 'e'.
'e'+'nn'+'ui'='ENNUI'
'in' is the link.
(Other definitions for ennui that I've seen before include "Boredom, tedium" , "bore" , "Sense of tedium" , "depression" , "Effect of boredom" .)