Supplements carefully as Doctor Kee's caught (4,3)

'supplements' is the definition.
(eking out is a kind of supplementing)

'carefully as doctor kee's caught' is the wordplay.
I cannot quite understand how this works, but
'as' could be 'so' (I've seen this in other clues) and 'so' is located in the answer.
an anagram of 'kee' is 'eke' which is present in the answer.
The remaining letters 'ut' is a valid word which might be clued in a way I don't understand.
This may be the basis of the clue (or it may be nonsense).

Can you help me to learn more?