While researching for this question, I found more details about what could be this (possible) adventure. Necho II hired a fleet of Phoenicians, who supposedly sailed from the Red Sea around Africa back to the mouth of the Nile in in three years.
The voyage was related by Herodotus as a complete circumnavigation of Africa in his History:
According to Herodotus, Necho II ordered a Phoenician-crewed fleet to leave Egypt from the east by way of the Gulf of Suez and to return via the Straits of Gibraltar at the Mediterranean's western mouth. Hence, he expected this expedition to navigate around Africa counterclockwise [sic, actually this was clockwise]. This would be a long journey, in which the crew would help support themselves by establishing temporary settlements on land where they would cultivate crops during the voyage.
According to the story, after two full years the fleet eventually rounded the Pillars of Hercules (the Straits of Gilbraltar), and returned to Egypt during the course of the third year.
Besides Herodotus story, there is no evidence if this travel. According to Egyptologist Alan B. Lloyd, the voyage was quite unlikely, since:
Given the context of Egyptian thought, economic life, and military interests, it is impossible for one to imagine what stimulus could have motivated Necho in such a scheme and if we cannot provide a reason which is sound within Egyptian terms of reference, then we have good reason to doubt the historicity of the entire episode.
This site has a possible reconstruction of the voyage. Another discussion is this reddit thread, where the answerer mention Ciaran Branigan as a supporter of the actual existence of this travel, while being skeptical about the voyage due to the same reasons Alan Lloyd did, and the fact "Ptolemy's account of the world which contained all of the Western knowledge of geography from c. 150 AD, considered Africa as an endless Southern land mass", impossible to circumnavigate, in contradiction with this supposed journey.
However, these sources did not mention where comes the name "Hamon". But they mention a Carthaginian navigator, Hanno, who did another travel to Africa, now starting from the West, sailing to Gibraltar. According to Pliny the Elder, Hanno actually managed to circumnavigate Africa, but modern accounts consider they only got to the West African coast, in some point between Senegal or Gabon; the supposed primary source for this journey (a tablet which was later translated to Greek) ended with "For we did not sail any further, because our provisions were running short". It should be noted that the tablet was deposited in the temple of Baal Hammon, also know as Hamon, the chief god of Carthage. So, it's possible that the book confounded these two travels (since this last one was not related with the Egyptians).