なおす is the most plain, and also natural word when to make something correct, fix or cure. In other words, getting something to be its right/natural state. Note that we use various Kanjis for なおす to make it clear how to do so. ex. correct or fix -> 直す; cure->治す 正す/ただす/ is often used with some certain nouns, like 姿勢/しせい/を正す (adjust posture)、不正/ふせい/ を正す (right a wrong). 矯める is a big word, we rarely see.

What verb is right for referring to 'correcting' someone's work? ※If "work " here is used "sentences", my answer is like the following. casual ぼくの／わたしの　にほんごを　みて／チェックして（ください）。 Bokuno/Watashino Nihongo o mite/chekkushite（kudasai）. Watch/Check my Japanese（please）. usual わたしの　にほんごを　なおして　ください。 Watashi no Nihongo o naoshite kudasai. Please correct my Japanese. polite わたしの　にほんごを　なおして／ていせい<訂正>してformal　くれませんか。 Watashi no Nihongo o naoshite/teiseishite kuremasenka? Could you correct my Japanese?　 more polite わたしの　にほんごを　なおして／ていせいして／てんさく<添削>してstiff　いただけませんか。 Watashi no Nihongo o naoshite/ teiseishite/tensakushite itadakemasenka? Could you correct my Japanese, please?

Maybe, would you like to use this word to ask others to correct your notebook entry? If so, you should ask about the whole sentence, such as 'Could you correct this entry?' This word has a lot of Japanese equivalents as bleh are referring to some: http://ejje.weblio.jp/content/correct