How do you influence an RFP when you hear about it late? Is there a way to still win the opportunity?

How to influence in general? Act as a trusted advisor that makes use of the RFP to demonstrate the superiority of your product and educate the customer. Try to get a hand on the material before it gets published and influence that your USPs gets mentioned so only your product fits the requirements. Make use of partners, advisors that helped the prospect/customers with the RFP.

When you feel that you get the RFP late? Be bold enough to ask for the reasons. They ask you to make a bid so you are also free to make demands on information. If they are hesitant with giving you reasons or you obviously feel that they are not telling you the truth - it is possible that they only need you as a counteroffer or to have reference bids. If you feel that no matter what you will propose they won't choose you - save the effort. But try to convince them that a RFP is always a possibility to challenge the status quo, research the market for better solutions and go with a product that is better than the current one (if there is any).