How do you post a partial payment against an invoice and send a sales receipt?

I believe that you only have to do a sales receipt, as that records the payment. 

Create a sales receipt to record the payment you received. Send the transaction by clicking the Save and send button. Let me show you how:

1. Go to the Plus icon and select Sales Receipt.
2. Choose your customer.
3. Enter all necessary information, including the payment amount, even if partial. 
4. Hit Save and send.