Interestingly, I worked at a secure government funded facility for years, and they had an incidental use policy that allowed some personal usage on government owned computers. This did not include running an outside business, day trading, or doing anything for profit, but it might include sending an email to your kid's soccer team or getting a grocery list from a spouse. Of course, all of the normal work use restrictions still applied (adult content etc..). At this facility, I basically assumed that every keystroke was monitored and tracked, which they had a perfect right to do on their computers and servers. So, if I ever did use my computer for personal, I thought of it like the cyber security department head was in my office looking over my shoulder. Never had a problem.
BTW - Eventually, the facility allowed folks to bring their own devices to the office. Mainly, folks brought personal iPhones and iPads, since they didn't plan on doing anything of a personal nature that was very time consuming or required a personal laptop whilst at work.