Hurry round capital city, costing sleep (9)
I believe the answer is:
hibernate
'sleep' is the definition.
(hibernating is a kind of sleeping)
'hurry round capital city costing' is the wordplay.
'hurry' becomes 'hie' (historical word meaning go quickly or hasten).
'round' means one lot of letters goes inside another.
'capital city' becomes 'bern' (capital city of Switzerland).
'costing' becomes 'at' (as in a shop stocking items 'at' a certain price).
'bern'+'at'='bernat'
'hie' going around 'bernat' is 'HIBERNATE'.
(Other definitions for hibernate that I've seen before include "Breathe in for long winter sleep" , "Pass the winter in a dormant state" , "go out in the snow?" , "Breathe in to settle down for winter sleep" , "Become dormant" .)