The following is from The Catholic Biblical Association’s book, A Commentary on the New Testament, published in 1942, page 68, concerning Matthew 8:20:
20. This is the first time in this Gospel that the expression Son of Man is used. Our Lord used it often in speaking of Himself. One reason for this was probably the Semitic custom of polite speech which discouraged the too frequent use of the first person singular “I”]. But the reason why Jesus chose this specific title for Himself was because it was definitely Messianic yet at the same time not clearly understood by the people as a synonym for Messias [Messiah]. An examination of the numerous places where it is used (not only in the Synoptic Gospels [Matthew, Mark, and Luke] but also in John) shows that Jesus used it of Himself in His office of Messias but with special emphasis on His human nature. No doubt it is ultimately based on Dan. *7,*13-14 [Daniel 7:13-14], for even the enemies of Jesus understood this passage as Messianic (cf. Matt. 26, 64 ff; Mark 14, 62 ff). But that it was ordinarily not understood as such by the people seems clear from John 12, 34. Therefore by calling Himself the Son of Man Jesus both proclaimed Himself the Messias and at the same time concealed this truth from the people because He did not wish to be taken for such a Messias as the people were expecting. (See Commentary on [Matthew] 8, 4). Since this humble title seemed to stress the humanity of Jesus a the expense of His divinity, with the exception of Acts 7, 56, where there is reference to the prophecy of Dan. 7, 13, it is never used in the New Testament by any one but Jesus Himself.
The above mentioned Commentary on Matthew 8:4, page 64-65, says in part:
4. See thou tell no one: most of the miracles of Jesus were not done primarily to prove to His immediate audience that He was the Messias and the Son of God but rather out of sympathy for the afflicted. On the other hand, during most of His ministry, at least in Galilee, He endeavored to conceal His miracles. For He did not wish to be proclaimed the Messias as long as the people had a false idea of the Messias as a mighty temporal prince (cf. John 6, 14 f). For similar instances of our Lord’s efforts to conceal His true nature until the proper time cf. Matt. 9, 30; 12, 16; Mark 1, 34; 3, 12; 5, 43; 7, 36; Luke 4, 41. …