Newspapers (and other journalists) often assign reporters to a "beat," which is an area of focus.
The person who has that beat is responsible for making sure that they get all of the newsworthy items in the area. For example, if you were on the beat for the local soccer team, you would need to know all the coaches, all the players. You would know if anyone was injured. You would have relationships with the people who worked at the stadium. You would follow any rumors about the team - and have people to call to confirm them. You attend all the official press conferences, and you set up your own interviews with people after the fact.
And if another newspaper found out information first, you would get blamed - because your job is to know everything about that team.
Of course, the team knows who is working their beat, and they try to feed those people favorable stories. While the reporter tries to stay objective and stay on the team's good side (you don't want the team to stop talking to you, but you also don't want to lie for them - or even not tell the truth about them when it's a hurtful truth)
So, the "Trump Beat" are the people who followed Trump around from place to place and to report on him. This reporter took a gift (bribe) from Trump (tickets) which is a huge ethical problem.