Which one is correct?
You heard it correct
You heard it correctly
Does the same apply to read it correct[ly]?
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Correct (an adverb is needed to modify the verb heard or read):
You heard it correctly. / You read it correctly.
What may be throwing you off is the common expression "heard you right":
If I heard you right, we are going to need more money.
In that phrase, "right" is an adverb, even though it sounds like an adjective.
The standard way of saying this is
You heard it correctly.
There is a trend in many dialects of American English to allow the use of adjectives for adverbs in certain circumstances. This is one of those circumstances. If you think you're hearing "heard it correct" a lot from Americans, you are probably right.
If you click through, Google shows you roughly 500 hits for "heard it correct" and "read it correct", just as many as for "heard it correctly" and "read it correctly", and I presume that there are lots more hits for each of these phrases that they're not showing you. So this "mistake" is very common, although I am sure that "heard it correctly" is used much more frequently.
I wouldn't advise trying this at home unless you know how to do it right, because you may end up sounding absolutely terrible, even to American ears:
*He goes bold where no man has gone before.