The press more often than not focuses on animals well-known.
this sentence is so tricky that I do not understand why the word "than" stand in this position. And the meaning make me confused too.
the press more often than not focuses on animals well-known
Replace "more often than not" with the word "usually" and you'll get the meaning.
It'll then be
the press usually focuses on animals well-known
Read more on ODO.
Or, as Hank suggests, you could also rearrange "well-known" this way
the press usually focuses on well-known animals