Let's say you are using an API gateway like Apigee or Amazon API Gateway and your public address for your API is http://my.public.dns.com/path/v1/articles. This gets routed through your gateway to an internal host http://some.internal.host.com/v1/articles. Now if your internal API returns relative links to itself then when they get served to the client they will be incorrect as they are based on its path not the actual public path. I know I can resolve this by transforming the response using the tools available in the respective gateway.
The question I have is; is there a standard way for a gateway to communicate the public path to the downstream component? I was thinking there might be a similar HTTP header to X-Forwarded-For.
There is no standard similar header.
Note that I think you mean if the back-end returns absolute links. If the links were relative, they'd already be correct. Relative links use no leading / thus are "relative" to the current directory -- whatever it may be - and so any external prefix is retained by the browser and the links transparently remain valid. Relative links can also backtrack a directory level with a ../ path prefix for each level.
Note also that API Gateway doesn't require any path prefix if you're using a custom domain name for your API This necessarily limits you to deploying a single "stage," but that's a reasonable tradeoff for a more flexible path... so the easiest solution might be to use a path in your API matching those internal paths.