“The etymology of Mykenai from mykes, although frequently proposed, was rejected by Krahe (Gnomon 17, 1945, 472). There seems, however, no reason why the city's name is not correctly associated with mykes. The declensional stem of mykes alternates between myket- (third declension) and myke- (first declension); the lack of -t-in Mykenai is therefore not significant. For a city related to a plant, cf. Mekone from the opium poppy mekon. Mykenai, furthermore, is a feminine plural, like Thebes (Thebai) and Athens (Athenai); as in the names of those cities, Mykenai derived from the nymph of the place, Mykene (Homer, Odyssey 2. 120; Hesiod, frg. 246, Merkelbach and West; etc.), the primordial bride whose descent into death there established, or continually re-established, the compact with the chthonic realm upon which union. the living city could be founded. There was also a tradition about an eponymous male founder, Mykeneus, whose father was Sparton, the 'sown man,' apparently an autochthonous inhabitant like the Spartoe at Thebes growing from the land (Acusilaus, frg. 16, Jacoby Eustathius on Homer, Iliad 2. 569 p. 289. 47; cf. scholia to Euripides, Orestes 1239). It is formulaic in foundation myths for the narrative to show a pattern of mediation between two versions of origin, autochthony and in migration; the autochthonous inhabitant quite naturally displays botanic characteristics and the mediation between opposite claimants to the place is effected via the sacred marriage. The name of the city Mykenai, therefore, would suggest that the mykes figured in Mycenaean religion. There was a tradition that at the neighboring city of Corinth, the original inhabitants were mushrooms that Sisyphus converted into men (Ovid, Metamorphoses 7.392-3; cf. Apollodorus 1. 9. 3).”
However I do not see here any convincing argument here.
While it is true that there was an alternative stem mukaa- for mukaat- (though attested only in the plural), deriving a female name as Mukaanaa from mukaa- is not a normal derivation in Greek.
In my opinion, the most likely is that the woman name Mukaanaa and the city name Mukaanai were indeed pre-Greek, so of completely unknown meaning.
A millennium later, the Greeks did not know more than us about what Mycenae might mean, so, as it always happens in such cases, they also thought about fanciful etymologies of the word, e.g. that it might be related to mushrooms, and they introduced such elements in their legends.
“The etymology of Mykenai from mykes, although frequently proposed, was rejected by Krahe (Gnomon 17, 1945, 472). There seems, however, no reason why the city's name is not correctly associated with mykes. The declensional stem of mykes alternates between myket- (third declension) and myke- (first declension); the lack of -t-in Mykenai is therefore not significant. For a city related to a plant, cf. Mekone from the opium poppy mekon. Mykenai, furthermore, is a feminine plural, like Thebes (Thebai) and Athens (Athenai); as in the names of those cities, Mykenai derived from the nymph of the place, Mykene (Homer, Odyssey 2. 120; Hesiod, frg. 246, Merkelbach and West; etc.), the primordial bride whose descent into death there established, or continually re-established, the compact with the chthonic realm upon which union. the living city could be founded. There was also a tradition about an eponymous male founder, Mykeneus, whose father was Sparton, the 'sown man,' apparently an autochthonous inhabitant like the Spartoe at Thebes growing from the land (Acusilaus, frg. 16, Jacoby Eustathius on Homer, Iliad 2. 569 p. 289. 47; cf. scholia to Euripides, Orestes 1239). It is formulaic in foundation myths for the narrative to show a pattern of mediation between two versions of origin, autochthony and in migration; the autochthonous inhabitant quite naturally displays botanic characteristics and the mediation between opposite claimants to the place is effected via the sacred marriage. The name of the city Mykenai, therefore, would suggest that the mykes figured in Mycenaean religion. There was a tradition that at the neighboring city of Corinth, the original inhabitants were mushrooms that Sisyphus converted into men (Ovid, Metamorphoses 7.392-3; cf. Apollodorus 1. 9. 3).”