This is a two step process:
1) Find the name of the map file the RouteGadget is using.
Right click on the browser - it seems to work best over on the right hand side where all the RG controls are. Choose the command "View Source" or words to that effect.
You will then see a new browser window full of text. Use ctrl-F to bring up the text search and search for "/kartat/". Some text that looks a bit like this should show up:
var imageUrl = '../kartat/43.jpg'+mapidentifier,
This shows us the name of the map file (43.jpg in the example).
2) Create the complete url (web-address) for the map file
Your browser will already be showing you the address for the RouteGadget - something like:
Just take the first part of that (the bit before /cgi-bin/ ) and stick the text "/kartat/" followed by the map file name on the end.
That will give you the full url. In this case its: