There are many reasons of interest in this passage of the Homeric Hymn to Apollo. It brings out the important role played by poetry in the context of religious celebrations, in this particular case in honour of Apollo. Poetry is the main place where human and divine encounter each other. It is significant that the god to whom the poem is addressed by the poet is the Lord Far-shooter, even called Apollo musagetes (i.e. the leader of the Muses), the god of poetry always portrayed with his faithful dual tool: the bow/ lyre ("I want the lyre and the crooked bow as my things.", line 131)  . The fact that the poet dedicates a poem to the god of poetry, to nothing less than the personification of Poetry, allows us to distinguish different levels of analysis and even to engage a meta-poetical discourse, i.e. a thought by the poetry on the poetry itself.
Firstly, you have the occasion, the reason for which the author wrote the poem. This hymn was performed to celebrate a combined Delian and Pythian festival on Delos in honour of Apollo, decided by Polycrates in 523 BC.
Secondly, that is the frame in which the real content of the poem is set, the direct speech of the author appealing the goddess Leto and her son Apollo. In the opening you have the poet's worrying about the difficulty of the task of singing of such a great and worshipful god, repeated in the middle of the hymn, while in the last two lines of the composition you see the poet taking leave of the god.
Thirdly, the mere celebrative content of the hymn, namely the stories of Apollo's birth in Delos, of his feats seeking the fit place for his oracle and of the settling of it in Delphi, with Cretan men as ministers.
Lastly, in the middle of the composition you find the passage I am going to analyse, where two specular scenes are portrayed by the poet. The former shows a Delian celebration where, boxing and dancing at the charming sound of the Maidens of Delos' hymning, Ionians are gathered to celebrate Apollo. The latter takes place on Olympus where you see Apollo himself delighting the congregation of all the immortals with the sound of his lyre together with the Muses'singing about gods and humans.
It's worth to notice the way the two sights are linked. Between the earthly looking at the divine and the divine looking at the earthly, there is again the poet himself in first person, a mortal as he is, addressing Apollo, the immortal god of poetry, through his servants, the Maidens of Delos. He is asking for Apollo's favour and for his authority to certify his fame of poet in the name of the god; in return for this service, he promises the Maidens, that are the mean of the god on earth, to spread their reputation (and together whit their even the god's one) among humans.
As you note thanks to the great recurrence of words belonging to the semantic field of persistence  , poetry concerns fame (kleos), and fame is strictly related to the idea of remembrance through time and therefore of eternity. In short, poetry is the place of everlasting fame and bridges the gap between the man's temporal dimension and the gods' timeless one.
In this work I shall focus on a particular function of poetry that concerns the dual feature of the relationship between man and god brought out by this passage: the poetry as a mutual gift, from man to gods as a celebrating tribute, and from gods to man as a divinely inspired craft. This relationship shall draw attention to the topics of eternalizing poetry and (im)mortality.
Poetry, in the form of hymn, is the way in which men celebrate the deities: it is a prayer, the greatest tribute one could address them.
In the meanwhile, at the basis of ancient poetry lies the concept of enthusiasmos  , that means a kind of possession of the poet by the god. That is to say that the poetical inspiration comes from the god as a gift.
This is the reason because of Socrates, in Plato's Ion, dismisses the poet's claim of knowledge: the poet is the mere instrument used by the god, an empty and unaware speaker. But in this sense poetry is also a grace and the poet is chosen by the god: it is a privilege for him to be bard of the god.
A fundamental consequence of this gift is that through poetry the poet can reach the poetical fame that can grant him a sort of immortality. Critics and poets themselves, in the past as well as in more recent times, have defined this feature the eternalizing power of poetry  . In lines 189-193 the author tells us that at the Zeus' court the Muses answer to the music of Apollo's lyre singing "of the gods' divine gifts and of human sufferings - all they have from the immortal gods and yet live witless and helpless, unable to find a remedy for death or a defence against old age." In my opinion the content of the singing of the Muses is quite ironical since it follows immediately the poet's demand to the Delian Maidens for a lasting renown: actually, right that one is the remedy men found to endure the death.
The literary immortality of the poet is exactly the same that he grant the god whom he is hymning to. In my opinion, this is due to the fact that the task of the poet is not merely a passive practice. It involves in same way a work of composition: a creation. When the divinely inspired poet elaborates the hymn he has materially to shape the poem. It would seem banal but we have to keep in mind that inspired or not a hymn, a poem, is always made by words.
The words of poetry make, in a culture, the place the gods need to exist. Gods do are immortal but only until someone cherishes any memory of them. Greek gods have been, are and, cultural catastrophes excepted, will be immortal because, thanks to poetry, people know them. But what whether we would have no evidences of their existence? And what before their first representation? W hat whether no account of a disappeared people has not reached us? In order to approach these questions should be useful to considerate what R. Hunter states about the ontology of gods.
In his account a god has no existence beyond the borders of literature and the related collective conscience of a people; a god is a patchwork, the artificial amount of all the attributes ascribed to him by the tradition. According to this a god's existence lasts as long as his cult and its memory last. Gods have no age and live in a timeless dimension, that otherwise is itself temporal: they 'are born' with their first epiphany and they 'die' when they are forgotten. Gods live in an indefinite mythical past that is made present every time a poem or a celebration is performed, and that is that, no longer.
Consequently, gods live an imaginary, artificial life and they live thanks to men's believes; they have the illusion of a powerful unaging and immortal existence but that is just what men's believes accord to them, what men want (or need  ) them to be.
While the poet of the Hymn to Apollo, in lines 171- 178, promises to spread the reputation of the Maidens among "the well-ordered cities of man" and to not to "cease from hymning the far-shooter Apollo of the silver bow", he is guaranteeing to the god the immortality itself from which he derives his powerful and worshipful authority: authority that only is capable to qualify the poet for the fame than wins the temporal death.
This argument makes clear why is significant the position of the poet to link the two parallel scenes of celebrations, the mortals' and the immortals' ones. In the relationship between men and deities the real pivot is not Apollo, the immortal god of poetry that give poetry to the mortals, but the mortal poet that gives life to the god of poetry to get his kleos aiei.