New York — The chief executive CNN pushed out of a job on Wednesday faced mounting problems in his first year leading the struggling network: viewership and profits were declining, programming blunders were growing and the network’s journalists were losing confidence by the day. Chris Licht’s very bad year culminated in a damning magazine profile last week, and just a few days later his tumultuous 13-month tenure was over. Licht, 51, was informed of his ouster Wednesday morning, and it was announced to the staff at the daily editorial meeting — the same place where Licht had said two days earlier that he would “fight like hell” to earn the trust of those around him.