Can I file a claim against a government agency for making false statements and emotional distress?

I won't say it's impossible, but this kind of claim is very unlikely to succeed. Government agencies have extra levels of protection that individuals and corporations do not have. If an action is within their "discretion" then they can't be held liable, even if what they did is an abuse of discretion. Then there are lots of technical issues about the extent to which damages can be claimed, or proven. Possibly, you may be able to get more records, if that is what you are looking to do, depending on what you want, why you want it, and why they are denying the request.