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.