Can anything be done to fix my puffy nipples/ (Photos)

Unfortunately there is no non-surgical treatment for gynecomastia, and so it does require a small operation.  It is a very embarrassing condition for many males and a lot will go through their whole life feeling self-conscious and thinking that nothing can be done to correct it.  My advice would be to research board-certified plastic surgeons in your area who have a lot of experience in this.  You will learn more about the expectation of results, cost of surgery, etc.  There are financing companies that will allow someone to finance surgery for another individual, and there are other options as well.  Even if you end up having to wait until you can save for the surgery yourself, just be patient and comforted in the knowledge that this is very fixable.  Best of luck!

Hello and thank you for providing photo reference. 
As per your photograph it does seem you have gynaecomastia and that you would need gynaecomastia surgery in order to remove any excess glandular/breast tissue. Please consult with a board certified plastic surgeon who specializes in this surgery for more accurate and personalized assessment. Compare cost with various surgeons in your area and ask if they do business with finance companies and compare their rates as well. 
In hopes this has helped answer your question. 
Best Regards