Of these two words, which one is correct: single handedly (no hyphen) or single-handedly (hyphen)?

Use a hyphen when combining these two words into one. That’s because together the two words act as one part of speech. Typically, single is an adjective, but in this case it works as an adverb. To wit, The brave starship captain single-handedly defeated the alien warrior.

A note that in British English, single-handed is preferred while American English tends to add the ly and more frequently uses single-handedly.

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My name is Rob Bignell. I’m an affordable, professional editor who runs Inventing Reality Editing Service, which meets the manuscript needs of writers both new and published. I also offer a variety of self-publishing services. During the past decade, I’ve helped more than 300 novelists and nonfiction authors obtain their publishing dreams at reasonable prices. I’m also the author of the 7 Minutes a Day… writing guidebooks, four nonfiction hiking guidebook series, and the literary novel Windmill. Several of my short stories in the literary and science fiction genres also have been published.