What is the give and take relationship between media and public officials?
I would say that the major give and take here is that both sides try to use the other for their own benefit. The public officials try to use the media to portray themselves and their ideas in the best possible light. The media try to use the public officials to get stories that will sell. Neither side's goal is exactly compatible with the other so there is a bit of give and take.
For a public official, it would be great if the media would just print what the official says. It would be great if the TV stations would broadcast the pictures that the official wants shown. This would make the official look good. So the official has to curry favor with the media to try to make this happen.
For the media, what the public official wants published is not always that interesting. The media would rather publish stuff about scandals, for example. But the media also needs inside information from the officials so that their stories can sound official and so they can get little facts that they might use against various public officials. So the media has to curry favor with the officials.
So both sides are trying to curry favor with the other while, at the same time, trying to manipulate the other side for their own benefit.