You might post how you are accessing the site. There are ways to decline requests in all formats as far as I'm aware.
Decline game request
I acces my games when I get notification that it is my move. However at some stage another member, who I did not know, assumed I wanted to start another game. I have been trying to notify him that I have enough games and do not want to start another game by sending messages but it has not worked. I beleive I have set my settings up so that I do not receive any more requests but it does not seem to be working. I keep automatically being set to "on vacation" because the unwanted game is running out of time. How do I remove this unwanted game?

A challenge that isn't accepted should not cause vacation. If that is happening, you should open a ticket instead of a forum post.
That said, are you on the website via a computer, on a mobile browser, on the app? I don't get many challenges so I can't recall right off the easiest place to decline them but can find out. Though, I would prefer not to search that out for each access method.
I access via the web site www.chess.com on my desktop computer and most of the time log in to access my games, other times I just access via the automatic email notification from chess.com subject is:
"Chess.com - Your Games Have Been Paused" content:
The games which triggered this event are:
CLICK HERE to view your games!
This leads me to a game from member soumya_kanti_sahu. I have been trying to decline this game but have not been able to. I have not accepted this challenge but it keeps reappearing and all of my games are automatically paused and I am set as "on vacation". All my games have a 14 day "next move time" except for this one which appears to be 3 days.
I haven't declined any requests, I haven't been able to find out how. Initially I merely ignore the challenge then my games were paused (On Vacation) automatically by the website whenever this particular game reached a time limit of 30 minutes remaining. The full text of the automatic email is:
You may have as little as 30 minutes remaining in one or more of your Online Chess games and because you are a Premium Member your account has been automatically set to "Vacation". Please login soon and continue your games!

It this is the game, you may have accidentally accepted the challenge.
http://www.chess.com/daily/game/124105914
If you hadn't, I wouldn't see it. Just resign/abort. No moves have been made so it won't impact your rating.
There does not seem to be any mechanism for declining requests to enter into a game that I can find. How do I decline, do I just add a note to the game request or is there a more formal mechanism for decline requests to start a chess game?