University originally paying part of castle’s running costs (6)
I believe the answer is:
upkeep
'running costs' is the definition.
(maintenance costs)
'university originally paying part of castle's' is the wordplay.
'university' becomes 'u' (abbreviation for university).
'originally' suggests taking the first letters.
'part of castle' becomes 'keep' ('keep' is part of 'castle').
The initial letter of 'paying' is 'p'.
'u'+'p'+'keep'='UPKEEP'
(Other definitions for upkeep that I've seen before include "Looking after" , "Running costs" , "Maintenance costs" , "Cost of support" , "Cost of maintenance, say household" .)