When I click that link (the one that used to be displayed in your post), this is the response:
{"code":401,"error":"Authorization required","message":"You must have an API key to use this resource."}
As an experiment, add apikey
to the second URL in the same way it’s used as in the first URL.
EDIT
This seems to be more of an API access problem (composing the correct URL) than a Home Assistant problem. I’m unfamiliar with mobygames API but, for many APIs, it’s often required to include your API key in each individual request.