Ulysses by Charles Lamb

Lovecraft wrote, "Then a tiny book in the private library of my elder aunt — the story of the Odyssey in "Harper's Half-hour Series"— caught my attention ..."

Well, Jacket Flap has a scanned copy with an excerpt if you would want to buy it. Here is the excerpt. (This is the edition of Harper's Half-Hour Series #92).
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Excerpt from book {with typos}:

to them a thousand times; and full in his view stood Mount Nerytus with all his woods: so that now he knew for a certainty that he was arrived in his own country, and with the delight which he felt he could not forbear stooping down and kissing the soil.

CHAPTER VIII. The change from a king to a leggar.- Eumceits and the herdsmen.- Ttlemachus. , Not long did Mineiva suffer him to indulge vain transports, but briefly recounting to him the events which had taken place in Ithaca during his absence, she shewed him that his way to his wife and throne did not lie so open, but that before he were reinstated in the secure possession of them, he must encounter many difficulties. His palace, wanting its king, was become the resort of insolent and imperious men, the chief nobility of Ithaca and of the neighbouring isles, who, in the confidence of Ulysses being dead, came as suitors to F2 Penelope. The queen (it was true) continued single, but was little better than a state-prisoner in the power of these men, who under a pretence of waiting her decision, occupied the king's house, rather as owners than guests, lording and domineering at their pleasure, profaning the palace, and wasting the royal substance, with their feasts and mad riots. Moreover the goddess told him how fearing the attempts of these lawless men upon the person of his young son Telemachus, she herself had put it into the heart of the prince, to go and seek his father in far countries; how in the shape of Mentor she had borne him company in his long search; which, though failing, as she meant it should fail, in its first object, had yet had this effect, that through hardships he had learned endurance, through experience he had gathered wisdom, and wherever his footsteps had been, he had left such memorials of his wo...