"Can I ask you all a huge favor. Can I first of all say a huge thank you for those of you that are here tonight," he said outside the theater's stage door. "Its been a hell of a week, it's been one damned thing after another."
Cumberbatch, 39, plays the titular character in the classic Shakespeare play. He spoke to fans about the distracting "red lights" of phone cameras in the audience and why filming of the production needs to stop.
"I can see cameras, I can see red lights in the auditorium, and it may not be any of you here that did that, but it's blindingly obvious," he said. "I could see a red light in about the third row on the right and it's mortifying. There's nothing that's less enjoyable or supportive as an actor on stage experiencing that."
The Oscar nominee asked fans to "Tweet, blog and hashtag the s---" out of his plea.
"I can't give you what I want to give you, which is a live performance that you'll remember, hopefully, in your minds and brains whether it's good, bad or indifferent, rather than on your phones," he said.
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Cumberbatch has received mixed reviews for his turn as the prince of Denmark in the Lyndsey Turner-directed show, from praises to pans.
The star's wife, Sophie Turner, was on hand to support him earlier this week.