About cult practices, we know very little, though we have quite a lot of material. The cave at Polis Bay was in use as a ritual site from Mycenaean times (Late Helladic III) up until the 1st century CE. But its association with Odysseus is most likely a consequence of the Odyssey, and therefore later than the epic (mid-7th century BCE). The only direct evidence linking it to Odysseus comes from right at the tail-end of the cave's use as a cult site.

It's been suggested that the twelve or thirteen 9th(?)-cent. cauldron-tripods found in the cave were the model for the tripods and cauldrons that Odysseus receives and hides in a 'cave of the Nymphs' in Odyssey book 13. There are of course always going to be 'Homer-realists' who insist that Homeric material goes back all the way to the Bronze Age, and therefore the tripods would supposedly be offerings to Odysseus -- indeed some scholars discuss with straight faces why they can't be the exact tripods that Odysseus left there(!) -- but equally the votive offerings in the cave also have items dedicated to Hera and Artemis.

It's just the one mask that has the dedication to Odysseus, by the way; it dates to the 2nd/1st century BCE. As well as the mask, there are three inscribed sherds of the Hellenistic period addressed to Nymphs -- again based on Odyssey 13 -- and supposedly a tablet with an inscription to Athena, Odysseus' patron goddess (but this was apparently one of the finds removed by Schliemann; I can't trace whether it still exists somewhere or if Schliemann flogged it).

However, the cave was stuffed with votive offerings, and most of the ones where a dedicatee can be identified are not dedicated to Odysseus.

Votives are a typical feature of Greek cult sites: offerings of an artwork or any artefact, really, placed by a private worshipper to gain divine favour. In the Polis Bay cave, these offerings include a Geometric-era terracotta sphinx, numerous female figurines dating to the 8th-4th centuries BCE, over a hundred masks -- including the one with the Odysseus inscription -- ; Hellenistic nymph reliefs; and so on.

But of the 100+ masks, only one is addressed to Odysseus, and it's on a mask of a female face; by contrast, about 30 of the masks represent Artemis. As Carla Antonaccio puts it (An archaeology of ancestors: tomb cult and hero cult in early Greece, 1995, pp. 153-154),

If it were not for the tripods, the ascription of the shrine to Odysseus would be without any support earlier than the Hellenistic period. ...

In the absence of clear evidence to the contrary, Odysseus is best regarded as a relative latecomer to the cave. The earliest deities that can be certainly associated with it are Hera and Athena; next are the Nymphs, and Odysseus last.

Sylvia Benton, who conducted the excavations at the Polis Bay cave in 1930-1932, points to a 3rd century inscription which she thinks suggests a festival of games for Odysseus on Ithaki, and argues the 9th century BCE tripods were prizes in these games. But a 3rd century inscription is terrible evidence for a supposed 9th century festival. Antonaccio gives a good run-down of the available evidence in her description of the cave (pp. 152-155).

Still, cult-sites to Odysseus did exist elsewhere. There was a hero-cult in Laconia -- albeit as 'Penelope's husband' more than as a hero in his own right -- and there are the barest hints in Stephanos Byzantinos and the Odyssey scholia of cult sites in northwest Greece at places called Bouneima and Trampya (both otherwise unknown), and supposedly an oracle of Odysseus in Aitolia (mentioned in a commentary on Lykophron).