Every Hawaiian scholar has a take on this though it's really not a "Hawaiian" issue per se. There's a couple of things that we need to consider when we think about this question. One, the way that Hawaiian scholars and writers for the last two hundred years have had to use the medium of English and the Judeo-Christian framework in order to even explain aspects of the old Hawaiian religious system. Two, how in the minds of non-Hawaiians Cook still remains a figure of importance while in the minds of most Hawaiians, Cook was not even the first European to set foot in Hawai'i. Historically, Vancouver was more important (and more beloved) than Cook for Hawaiians except for that the toxic seeds of depopulation began with Cook. However, Cook's arrival during the Makahiki did play into the existing internal politics of Hawaiian society particularly between the nobility and the sacred professional (kahuna) classes--and both exploited Cook's arrival. Third, Cook's own stature in Britain and the world could not accept that Cook was going to be killed at some point on his third voyage. His own men wanted to mutiny. The chiefs of Tonga, Tahiti and Samoa were preparing to wage war against him. But for Britain and British colonial settlers in Australia and New Zealand, Cook remained an icon of the English Enlightenment and of the rising sun of the British Empire. His "discoveries" paved the way for British claims over indigenous lands throughout the Pacific.
I will not deal with the latter two but I will deal with the first.
Within the Hawaiian religious system, there are concepts and terms that simply do not translate well into a Judeo-Christian framework nor translate well into English. Many Hawaiian concepts do fit into certain ideas of Hinduism, Taoism, and other Eastern philosophies but most Hawaiian scholars and writers until recently have largely written to and for audiences (including other Hawaiians) who grew up within that Western religious experience and had to use language (and at times adjust) to relate to that framework. Whether or not Lono was considered a "god" is one case where the answer reveals how Hawaiian religious system is being forced to be seen from that Judeo-Christian framework.
In Hawaiian thought, the right answer is that Hawaiians did venerate Cook as a god and at the same time did reject him as a god. Both answers are correct but singularly are incorrect. Two correct answers seemingly contrary of each other may seem odd but that's completely logical within the Hawaiian framework because both answers together form the correct perspective.
Hawaiians believed that the ali'i were manifestations of their akua (gods) in the same way certain trees, birds, winds, rains, etc were manifestations. They were manifestations of the same 'ano lani, the same eternal and mystical mana that illuminated the akua. But Hawaiians also knew that their ali'i were also mortal and although they commanded respect, they were only human and when they erred (hewa), they were spiritually polluted (haumia) and needed to be corrected or overthrown. The people invoked ke'ehi (rebellion). Many are the chiefs that were overthrown and their names no longer spoken on the lips despite they being manifestations of the akua. For Hawaiians, not having namesakes or having their names recited in genealogical chants was the ultimate death for when names are spoken, the bones live once more.
Ancient Hawaiians also believed in the concept of kino lau, similar to the Hindu belief of avatara, where a deity can "ride" within a certain animal or natural form, normally to interact within this realm. The concept of 'aumakua may be derived from this. The state fish, the Humuhumunukunukuapua'a for example is a kino lau of Kamapua'a. The kukui nut tree is another kino lau of Kamapua'a, except for the nuts. This does not mean that Hawaiians of old literally believed that Kamapua'a lived in the body of the Humuhumunukunukuapua'a, but he could "ride" in that form to interact with this world. The setting of these kino lau also had another effect: it re-enforced the kapu system's resource management. People would not over fish, over hunt or over harvest in the fear that in doing so, it would upset one or more of the akua.
Related to that is the Hawaiian concept of noho akua, where an akua could be invoked or ride within a human, again, to interact within this realm. For example, my family generations ago venerated Kiha Wahine (often translated into English as "Dragon or Lizard Woman") whose specialization was the ritual of haka noho akua, an act where females put on special masks and special regalia and went into a trance to invoke the spirit of the Dragon Lady, perhaps similar to the Tibetan Bon oracle ritual. With the masks and regalia, they become Kiha Wahine, they become the avatar or vessel of that female god. But they were still humans and everyone knew that.
The same with Cook. While the Makahiki was on-going, some though perhaps he was a manifestation of Lono or maybe a being used as a noho akua for Lono, but they also knew he was a man. In addition to that, the Hawaiian mentality on chiefs were that they were to be respected until they did hewa or wrong. Once they committed hewa, they had become haumia (unclean) and as mentioned before, Hawaiians were not afraid of putting chiefs in place. The divine status did not exempt them from becoming haumia. Quite the opposite. The people expected their chiefs to be kūpono (upright) because of their status. They should know better because they have that 'ano lani within them. When Cook failed to act kūpono and when Hawaiians had become aware of the cloak of death that was covering the land due to the new diseases, the Hawaiians invoked ke'ehi, the stamping of the foot and setting of rebellion to remove the haumia from the lands--as they had done with chiefs before. Cook had thought that as a god, he was divine and infallible. But infallibility has no concept within Hawaiian thought and divinity has its limits--as does human patience.
So yes, did some Hawaiians venerate him? Yes within the Hawaiian religious context. Did Hawaiians see him as a man? Yes, within the Hawaiian religious context. Both answers are simultaneously correct and both answers support each other within the Hawaiian religious context.