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Yes, you can do better. Once you know spades break, you should play a club to the nine. The key is to keep the ♥K as an entry to table and the ♣K as an entry back to hand (to repeat the club finesse).
If you are starting clubs low to the nine I dont think it matters whether you kept the ♥K or ♥A. In one case you cross back to the hand via the ♣K to repeat the finesse. In the other case, when you are already in the hand because you kept the ♥A, you cash the ♣K then refinesse.
In isolation - say playing this suit in matchpoints - starting with the king wins an extra trick when there is a stiff Q or J offside - but here where you only need 3 club tricks it doesnt lose the contract. I thought the only combination you'd lose out to starting low was stiff QJ offside but as you always have to cash the ♣K before refinessing you will still get 3 club tricks with QJ stiff offside.
So now Im not sure which hand we win the trick in at the first trick matters at all.