Why do police helicopters always fly in circles?
Police helicopters don't always fly in circles. I've often seen my local police helicopter hovering. There are a few reasons you might have seen police helicopters flying in circles:-


Not all types of helicopter can hover out of ground effect. It might be that the police helicopters in your area can't hover at the heights they work at, with the weight of their usual equipment.
Helicopters are safer if they're moving forwards than hovering. In the event of an engine failure, they can autorotate to land safely, but they need to have some airspeed to be able to do this. The required airspeed depends on height (see the chart in the linked question), so if your news helicopters are usually operated higher up than police helicopters, they may be able to hover safely. (This is quite likely, as police aircraft are exempt from the 500 ft rule and the 1000 ft rule, while news helicopters are not.)
Police helicopters often circle to search an area for a fugitive or casualty using thermal imaging cameras. News helicopters tend to either follow a vehicle or hover to get a steady camera shot of a news event.