In this, where would I stick the partial private key?
The private key is used to calculate the public key.
You said you already have the public key.
Not sure if this is the right section. Sorry if not.
How can I calculate the address if I have the public key and a private partial key?
Once you have the public key, that's all you need for calculating the address.
Okay let me explain the scenario.
I'm using vanitygen. I supply a public key to it and a prefix. It then finds the vanity address using said public key and provides me a partial private key. Sites like bitaddress.com can take that public key and the partial private key and determine the address. The public key itself is not enough; that returns a different address.