Little attempt made to engage small government department (8)
I believe the answer is:
ministry
'government department' is the definition.
(I know that ministry is a type of government department)
'little attempt made to engage small' is the wordplay.
I cannot quite understand how this works, but
'little' could be 'mini' and 'mini' is located in the answer.
'attempt' could be 'try' (synonyms) and 'try' is present in the answer.
'small' could be 's' (abbreviation - e.g. clothes size) and 's' is present in the answer.
This accounts for all the letters.
This explanation may well be incorrect...
Can you help me to learn more?
(Other definitions for ministry that I've seen before include "service" , "Government department" , "Priests' work" , "cloth?" , "Government building" .)