King on the rise and partly open (4)
I believe the answer is:
ajar
'partly open' is the definition.
(I've seen this before)
'king on the rise' is the wordplay.
'king' becomes 'raja' (historical term for an Indian king or prince).
'on the rise' says the letters should be written in reverse.
'raja' backwards is 'AJAR'.
'and' is the link.
(Other definitions for ajar that I've seen before include "Partially open" , "Part-open" , "Not to answer" , "partly opened" , "Not quite closed" .)