In British Library’s not above making awful errors (8)
I believe the answer is:
blunders
'awful errors' is the definition.
'in british library's not above' is the wordplay.
'in' means one lot of letters goes inside another.
'british library's' becomes 'bls' (abbreviation for British Library).
'not above' becomes 'under' (I've seen this before).
'bls' going around 'under' is 'BLUNDERS'.
'making' acts as a link.
(Other definitions for blunders that I've seen before include "Embarrassing mistakes" , "Clumsy mistakes or gaffes" , "Mistakes made through ignorance or carelessness" .)