context:
As he showed me around the house, his gait was a bit awkward, as if,
at some level, he was just not that interested in walking. I asked to
see his office, and he pointed out an unremarkable chamber off a back
hallway. He doesn’t get as much done there as he used to, he said;
recently, he has been most productive on flights, when he has a block
of hours away from email and all the people who hope for an audience
with him.
Why people say with instead of of or to? Because I think it's not like friends or companions that should use with, but like a follower (not on the same class).
Best Answer
It's as Stephie said in her answer.
Normally, I would post this definition of audience with as a comment (as a hint that you could find the answer on your own if Stephie hadn't posted her answer; or in our case, as additional information after Stephie posted her answer). But in this answer, I want to point out that with a right dictionary, you can solve a lot of similar questions on your own.
It's reasonable that you expected audience of because the more common sense of audience is "a group of people who watch, listen to, participate in, or read something". That makes it easy for us to expect audience of someone or something along the lines of his or its audience.
But! here is my tip, when you find a familiar word used in an unexpected manner: Don't guess.
It's better to look it up in dictionaries, because chances are it could be an uncommon sense of the word; it could also be an idiomatic use of the word (and English has lots of idiomatic phrases which quite often the meanings are not compositional--that is, we can't understand the meaning of the whole phrase by reading it word by word).
Guessing could give you the wrong idea about the phrase. Why? Because being a non-native speaker, our intuition may not be as reliable as we think or hope it to be, which is simply because we already have another set of intuitions in another language, our first language(s).
Here is another tip: try to include a word before or a word after the word in question if you can't find a definition that makes sense the first time.
"And which dictionary should I use?" you may ask. My answer would be: any dictionary that works for you. But personally, I'd recommend Macmillan Dictionary. I have no affiliation with the dictionary. It's just that after a few years, it's clear to me that Macmillan always keeps learners in mind, even though they don't include the word "Learner's" in the title of the dictionary.
Macmillan Dictionary has several advantages over other dictionaries. It's online. Its definitions of words are in plain and clear English, which make it easy to read and understand the definitions. It comes with pronunciation, both in BrE and AmE, both in phonetic transcription and audio. It includes almost all common idiomatic uses of words and phrases. But that's not all! There are more subtle features such as "red stars" indicating word frequency, links to synonyms and related words, and so on.
Finally, if all those tricks with dictionaries fail, come here and post your question on ELL.
Happy learning!