A project is billed at 100% of the contract amount and then this amount is reduced for some reason. Can I refund this reduced amount in CORE?

Yes, this can be achieved via a workaround. Here are the steps for refunding a part of the contract amount in CORE:

1. Open the Invoices screen and click Create New > Manual Invoice. Check Create Manual Invoices for details.
2. Create a negative manual invoice using as many items as needed to reverse the transactions involving the income accounts. 
3. You can view and process this invoice as final. This reduces the balance of the affected A/R and income account.
4. Now write a check for the client from the Checks screen using the Accounts tab. Check Create New Checks for details.
5. Make sure to reference the correct A/R account and project. This will increase the A/R account balance, canceling the effects of the negative invoice created in step 1.
   
   Now your transactions should be in balance ($0). This will leave the negative A/R invoice on your A/R report.