E.g. “What’s he think?”
Usually ’s is short for “is” so I don’t know.
The NOAD reports that 's is the informal contraction for:
It is certainly used in that way.
I don't regard the word "correct" as meaningful in this kind of question, so the answer to your question depends on what standard you choose to designate by "correct".
This can be a contraction of "what does", but I'd generally consider it a "colloquial contraction" insofar as it's a bit of a colloquialism to say "what's" instead of "what does". Basically, if you're describing someone who's said "what's" to mean "what does", or emulating their style of speech, then it's OK. Otherwise, it's a rather quaint contraction and I'd recommend against it.
As the NOAD reports cited above states, 's is the informal contraction for:
 is: "it's snowing" --this is the primary use. Here it would be pronounced /its/, even though "is" by itself is pronounced /iz/.  has: "he's gone" -- this is less common than the "is" meaning, but hardly rare. Since "has" is also usually pronounced /haz/, this 's would have a /z/ sound to distinguish it from 's = is.  us: "let's do it" -- less common than the "has" meaning, and not often used except with "let". "Us' has the /s/ sound, so "let's" would be pronounced /lets/.  does: "what's she do? -- not unknown, but not very common. Although "does" is usually pronounced /duz/, "what's" here would be pronounced /hwats/. (Like "it" in the first example above, "what" ends in an unvoiced plosive /t/--I believe that what "t" would be called in this case--so the following "s" would be an unvoiced /s/-sound, not a voiced /z/-sound. I hasten to add that I am not an expert in phonetics.)
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