The Chicago Manual of Style recommends against the use of ellipses to begin a quotation in running text (as opposed to block text). Because this is a style issue rather than a grammar one, you may receive several "correct" answers.
However, the importance of syntax should be made. Quoted material, especially in news reports, should be consistent with the message so the reader doesn't take things out of context.
Here are a few potential solutions:
"[I love the president, and] it's an honour to be here," said Anthony Scaramucci.
All General Jackson heard was "It's an honour to be here." But later, Colonel Stone said, "You missed the part where Anthony said he loved Trump."
However, given your example, there is no confusion Anthony supports and admires Trump in both his clauses, so you could legitimately use your second example without raising any hairs with an editor in an (American) publication. Why the writer would only include half the quote I don't really understand without more information from the OP.
"It's an honour to be here," said Anthony Scaramucci, President Trump's new White House communications director.