I apologise in advance if this is the wrong subreddit but i figured as I am searching for historical documents this would be a good place. During the Cuban Missile Crisis my great-grandfather was in correspondence with the Kennedy Administration (sending his letters from Australia) after responding to an advert in a Russo-American paper (or possibly Russo-Australian paper, details on that are foggy) for people with insider knowledge of the workings of the USSR. A small number of letters were sent back and forth between him and the administration. Time has caused the copies on this end of the line to be lost, would there be anywhere to contact or search for copies on the American side?
There are a couple of unknowns here that could make this tricky. For example, who was he corresponding with in the Kennedy Administration? From what you write, it sounds like he was applying for a job that didn´t really go anywhere. It also doesn´t sound like something that a key figure would have dealt with personally or necessarily saved.
You could try checking the National Archives in College Park, MD, which deal with correspondence and documents related to the Department of State.
There are individuals who supposedly look through archives on a contractual basis, in search of documents of personal significance like these, in and around the College Park area. I´d try and asking around in case you can´t come personally. Hell, it might be both less expensive as well as advantageous to have someone who knows their way around the archives.
The JFK library in Bostom, MA, is where JFK´s personal papers are kept, along with other documents from his administration... But from what you say it doesn´t sound like they would have your documents since your grandfather wasn´t directly corresponding with JFK himself.
I also recommend using the website of NARA at College Park to see what the professional archivists there recommend. JFK is a perennially researched topic, so I am sure that expert archivists there would be able to give awesome tips and advice specific to your situation. This is especially true if you add more details to your inquiry (when, with whom was he corresponding, more detail about the nature of the advert, etc.).
Good luck.
Edit: Also, on second thought, I'll take the opportunity to ping /u/caffarelli just in case, as our resident expert on archives, on the off chance I am leaving out another important place to look.