You can add/remove/override headers with an Integration Request Mapping Template.
In the API Gateway console, chose the relevant api/resourece/method. Go to Integration Request > Mapping Templates and chose your Content-Type (if requests are going to be received without a Content-Type header, set the Content-Type for the mapping template to application/json, which is the default behaviour).
Then in the actual mapping template add the following:
{
#set($context.requestOverride.header.apikey= "testMe")
}
This will add (or overwrite if it already exists) a header called apikey with the value "testMe" to all http requests downstream.
If you take this route, then you will need to also map over any other headers, path parameters, query parameters or body that you wish to pass through.
You could loop through the headers and query parameters like this.
## First set the header you are adding
#set($context.requestOverride.header.apikey= "testMe")
## Loop through all incoming headers and set them for downstream request
#foreach($param in $input.params().header.keySet())
#set($context.requestOverride.header[$param]= $input.params().header.get($param))
#if($foreach.hasNext) #end
#end
## Loop through all incoming query parameters and set them for downstream request
#foreach($param in $input.params().querystring.keySet())
#set($context.requestOverride.querystring[$param]= $input.params().querystring.get($param))
#if($foreach.hasNext) #end
#end
As you need to ensure that the header apikey is set to a default value, you should set the override for apikey before looping through the rest of the headers as only the first override will take effect.
The relevant AWS documentation can be found here.
The other alternative would be to point your API Gateway at a Lambda and make the call from the Lambda instead.