Frank started singing along, (much) to the annoyance of Ellen.
Is "much" necessary here or is it OK to leave it out?
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You can leave it out, much just stresses the intensity or degree of the emotion/feeling/state.
Similarly, much is optional in the following idioms:
(Much) to my horror, I looked up and saw the man standing in his yard.
(Much) to my surprise, they offered me a £4,000 scholarship.
I applied for the job, and, (much) to my amazement, I was hired.