Meaning - Gist
O AnjanEya! You are the foremost of the devotees; You have the privilege of experiencing the Supreme Bliss by always beholding the LordSrI rAma - the Consort of lakshmI, the King of ayOdhyA, the prop of everything and the Lord praised by this tyAgarAja!

At pre-dawn hours, the Lord takes bath of nectar holding Your hands; having offered the sacred foods to Lord ranganAtha, He partakes food laid by the hands of sItA; and then, He commands You to read out the holy scriptures.

Will it be possible for me too to have the privilege of beholding the Lord SrI rAma manifestedly and perform such a service at His Lotus Feet?

Word Division - Word-by-word meaning

pallavi
kalugunA pada nIraja sEva
gandha vAha tanaya

O AnjanEya - Son (tanaya) of the Wind God – one who carries (vAha) fragrance (gandha)! Will it be possible for me to have the privilege(kalugunA) of performing service (sEva) (like you) at the Lotus (nIraja) Feet (pada) of SrI rAma?

anupallavi
palumAru jUcucu brahmAnanduDai
paragE bhakta-agrEsara tanaku (kalugunA)

O AnjanEya - the foremost (agrEsara) of the devotees (bhakta) (bhaktAgrEsara) - who remains (paragE) as the one having the privilege of experiencing the Supreme Bliss (brahmAnanduDai) by always (palumAru) (literally often) beholding (jUcucu) the Lord!
will it be possible for me (tanaku) too to have the privilege of performing service (like you) at the Lotus Feet of SrI rAma?

caraNam
vEkuva jAmuna nI karamunanu-iDi
SrI kAntuDu-amRta snAnamu jEsi
pAkamulanu SrI ranga-ISuniki
arpaNamu jEsi tA
sItA karamulacE bhujinci ninu
sAtvIka purANa paThana sEyamanE
sAkEta patini sarva-AdhAruni
prAkaTamuga tyAgarAja nutuni kana (kalugunA)

O AnjanEya! will it be possible for me too to have the privilege of beholding (kana) manifestedly (prAkaTamuga) Lord SrI rAma - the Consort (kAntuDu) of lakshmI (SrI), the King (patini) of ayOdhyA (sAkEta), the prop (AdhAruni) of everything (sarva) (sarvAdhAruni) and the Lord praised (nutuni) by this tyAgarAja - who

at pre-dawn (vEkuva) hours (jAmuna), taking bath (snAnamu jEsi) of nectar (amRta) (kAntuDamRta) holding (iDi) Your (nI) hands (karamunanu) (karamunaniDi),

having offered (arpaNamu jEsi) the sacred foods (pAkamulanu) to Lord ranga nAtha (ISuniki) (rangESunikarpaNamu), Himself (tA) having partaken food (bhujinci) laid by the hands (karamulacE) of sItA, and

commands (anE) (literally say) You (ninu) to read out (paThana sEyumu) the holy (sAtvIka) (literally principle of light) scriptures (purANa)?

Notes

Variations - (Pathanthara)

References
⁴ – rangESa –

"Lord Ranganatha was first worshipped by Lord Brahma, then later given to Maharaja Iksvaku of the Raghu Dynasty, who brought the Deity to his capital of Ayodhya. Later, the Deity was worshipped by Lord Ramacandra. When Vibhisana visited Ayodhya he requested Ramacandra to give him the Deity so that he could worship Him at his capital in Lanka. Ramacandra agreed provided Vibhisana did not place the Deity on the ground on his return to Lanka – if he did so, the Deity would stay in that spot. Somehow or other, Vibhisana placed Lord Ranganatha down at Srirangam and that is where He has stayed ever since. However, unlike most Deities who face East, Ranganatha faces South towards Lanka to bless Vibhisana and his kingdom. It is said that Vibhisana returns every 12 years to perform worship of Lord Ranganatha."

Source - Sri Rangam - 1 and Sri Rangam - 2

The following verse in vAlmIki rAmAyaNa, uttara kANDa, Chapter 108 gives a clue to this episode –

kiMcAnyad vaktumicchAmi rAkshasEndra mahAbala |
ArAdhaya jagannAthaM ikshvAku kuladaivataM ||
ArAdhanIyaM aniSaM dEvairapi savAsavaiH |
tathEti pratijgrAha rAmavAkyaM vibhIshaNaH || (30,31)

SrI rAma said – “O best among rAkshasAs, this much more I wish to say. O mighty one, worship jagannAtha (vishNu) the presiding deity of ikshvAku family. Lord vishNu is to be worshipped always by even the Gods and indra.” Accepting the words of SrI rAma, vibhIshaNa replied “It shall be so”.

⁵ – sAtvIka purANa - Refer to 'Puranas - Type of'.
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Comments
¹ - kalugunA pada nIraja sEva – While translating pallavi alone and when connecting anupallavi word 'tanaku' – it would mean that SrI tyAgarAja aspires for similar kind of service at the feet of SrI rAma as performed by AnjanEya. However, when connected to the caraNa 'prAkaTamuga tyAgarAja nutuni kana', it may have two meanings – (1) SrI tyAgarAja aspires to behold SrI rAma manifestedly like AnjanEya or (2) SrI tyAgarAja aspires to witness the service rendered by AnjanEya (by changing the word ordering as 'SrI tyAgarAja nutuni pada nIraja sEva kana kalugunA). By adopting the first version, the pallavi wordings 'pada nIraja sEva' are rendered superfluous. By adopting the second version, the purpose of the word 'prAkaTamuga' (manifestedly) is defeated. Therefore, the first version may be appropriate.

² - vEkuva jAmuna - a period of three hours before dawn.

³ – amRta snAnamu – water is (also) called 'amRta' because it sustains life.
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kalugunA pada nIraja sEva - pUrNalalita