I get "thoughtlessly", "carelessly", "casually" and "lightly". I do not get "slightly". To me, "He speaks slightly" would mean he speaks only a very little bit, although it would be an odd way to say that.
"Slightly" is not often used to modify a verb, it usually modifies an adjective or a preposition, as: She was slightly drunk; they were slightly behind us; he is slightly better at chess.
Thinking further about this, I'd say you only use "slightly" with a verb when the verb has to do with measurable amounts: The ship deviated slightly from its course, my wage increased slightly, the level of interest varies slightly from day to day.
"Slightly" and "lightly" are the two primary meanings offered by Larousse for légèrement; although the sense of "thoughtlessly" is there also. http://www.larousse.com/en/dictionaries/french-english/l%C3%A9g%C3%A8rement/46493
Sorry. Let me explain better. If a word has multiple definitions, you can't substitute other words that are synonymous to one of the definitions NOT being used in that sentence. e.g. "He is sweet." One definition of sweet is sugary, but you can not say "He is sugary." Trying to use the sentence "He speaks slightly," is the same thing. It's using a synonym to ONE of the definitions that doesn't happen to apply to "Il parle légèrement." That's why it doesn't work. You can't use every definition for a word each time you see that word. The context is important.
Also if a french sentence doesn't use a word exactly the way we use it in English, that's okay. It looks like because "He speaks lightly" was an accepted answer, a lot of people thought "Well I wouldn't say 'He speaks lightly,' in English without more information following; it sounds weird..." Well a sentence can be strange in one language and be completely normal in another. It's like trying to explain the greeting "What's up?" to francophones, when the word-for-word translation is "What's above?" And when you translate "Ça va" word-for-word it means "It goes," which doesn't make any sense in English. When you learn a new language, sometimes you just have to accept the meaning in a new context and try your best to absorb it without worrying that it doesn't match your native language translation (Good advice my Montreal friend gave me when I was learning French there).
I entirely agree with all of that. It's what I try to get across to people when they complain that DL didn't accept something they randomly chose from those wretched drop-down hints. But I believe the main part of the discussion here is to actually grasp the French meaning and then how to express that in reasonable English. Which is what translation is about. Cheers!