Bishop Alan Clare, head of school, settles accounts (8)
I believe the answer is:
balances
'accounts' is the definition.
(balancing is a kind of accounting)
'bishop alan clare head of school' is the wordplay.
'bishop' becomes 'b' (abbreviation used in chess).
'clare' becomes 'ce' (I've seen this in other clues).
'head of' suggests taking the first letters.
The initial letter of 'school' is 's'.
'b'+'alan'+'ce'+'s'='BALANCES'
'settles' is the link.
I am not very happy about this link. It may belong to another bit of the clue.
Can you help me to learn more?
(Other definitions for balances that I've seen before include "Equalises, offsets" , "Keeps in equilibrium, like credits and debits" , "Maintains equilibrium like tightrope walker" , "Is poised; sums in account" , "Scientific instruments" .)