The "Hayagreeva Stotram" composed by Shri Vedanta Desika consists of 33 shlOkAs.
See a summary and analytical note of the composition.

The exploration here is to get at the meanings of each of the
words, investigate roots and related words.

Hayagreeva is often referred to in the invocative (sambodhanA) and
subject (dvitiyA) forms (vibhaktiH). The prosaical transformation
(anvayaH) is provided, where unobvious.

[1] "The Practical Sanskrit-English Dictionary" by Vaman Shivram Apte, Revised and Enlarged Edition, Motilal Banarsidass Pub
[2] "Shri Desika Stotra Mala" by Shri Ramadesikachariar Swamy, 2003 Edition, Lifco
[3] Shri V.S.Sadagopan's english translation
[4] "A Sanskrit English Dictionary", M. Monier-Williams, Motilal Banarsidass Publishers

[Invocatory Tanian on Swami Desikan by Kumara Varadacharya]

    shrImAn ve~nkaTanAthAryaH   kavi-tArkika-kesarI
    vedAntAcArya-varyo me       saMnidhattAm sadA hR^idi

[1.txt]

    j~nAnAnanda-mayaM devaM     nirmala-sphaTikA-kR^itim
    AdhAraM sarva-vidyAnAM      hayagrIvam upAsmahe

[02.txt]

    svatassiddhaM shuddha-sphaTika-maNi-bhUbhR^it-pratibhaTaM
      sudhAsadhrIcIbhirdyutibhiravadAta-tribhuvanam
    anantaistrayyantairanuvihita-heShAhalahaM
      hatAsheShAvadyaM hayavadanamIDImahi mahaH

[03.txt]

    samAhArassAmnAM pratipadamR^ichAM dhAma yajuShAM
      layaH pratyUhAnAM laharivitatirbodhajaladheH
    kathA-darpa-kShubhyat-kathakakula-kolahalabhavaM
      haratvantardhvAntaM hayavadana-heShA-halahalaH 

[04.txt]

    prAcI sandhyA kAcidantarnishAyAH
      praj~nAdR^iShTera~NjanashrIrapUrvA
    vaktrI vedAn bhAtu me vAji-vaktrA
      vAgIshAkhyA vAsudevasya mUrtiH 

[05.txt]

    vishuddha-vij~nAnaghana-svarUpaM
         vij~nAna-vishrANana-baddhadIkShaM
    dayAnidhiM dehabhR^itAM sharaNyaM
         devaM hayagrIvamahaM prapadye 

[06.txt]

    apauruSheyairapi vAkprapa~NcaiH
         adyApi te bhUtimadR^iShTapArAm
    stuvannahaM mugdha iti tvayaiva
         kAruNyato nAtha! kaTAkShaNIyaH 

[07.txt]

    dAkShiNyaramyA girishasya mUrtiH
         devI sarojAsanadharmapatnI
    vyAsAdayopi vyapadeshyavAcaH
         sphuranti sarve tava shaktileshaiH 

[08.txt]

    mandobhaviShyanniyataM viri~Nco
         vAcAM nidhe! va~Ncita-bhAgadheyaH
    daityApanItAn dayayaiva bhUyopi
         adhyApayiShyo nigamAn na cet tvam 

[09.txt]

    vitarkaDolAM vyavadhUya satve
        bR^ihaspatiM vartayase yatastvam
    tenaiva deva ! tridasheshvarANAm
         aspR^iShTa-DolAyitamAdhirAjyam 

[10.txt]

    agnau samiddhArciShi saptatantoH
         AtasthivAn mantramayaM sharIram
    akhaNDasAraIrhaviShAM pradAnaiH
         ApyAyanaM vyomasadAM vidhatse 

[11.txt]

    yanmUlamIdR^ik pratibhAti tattvaM
         yA mUlamAmnAya-mahAdrumANAm
    tattvena jAnanti vishuddha-sattvAH
         tvAmakSharAmakSharamAtR^ikAM tvAm

[12.txt]

    avyAkR^itAdvyAkR^itavAnasi tvaM
         nAmAni rUpANi ca yAni pUrvam
    shaMsanti teShAM caramAM pratiShThAM
         vAgIshvara! tvAM tvadupaj~navAcaH 

[13.txt]

    mugdhendu-niShyanda-vilobhanIyAM
         mUrtiM tavAnanda-sudhA-prasUtim
    vipashcitashcetasi bhAvayante
         velAmudArAmiva dugdhasindhoH

[14.txt]

    manogataM pashyati yaH sadA tvAM
         manIShiNAM mAnasa-rAjahaMsam
    svayaM purobhAva-vivAdabhAjaH
         kiMkurvate tasya giro yathArham 

[15.txt]

    api kShaNArdhaM kalayanti ye tvAM
         AplAvayantaM vishadairmayUkhaiH
    vAcAM pravAhairanivAritaiste 
         mandAkinIM mandayituM kShamante 

[16.txt]

    svAmin ! bhavaddhyAnasudhAbhiShekAt
         vahanti dhanyAH pulakAnubandham
    alakShite kvApi nirUDhamUlaM
         a~ngeShvivAnandathum a~nkurantam 

[17.txt]

    svAmin ! pratIcA hR^idayena dhanyAH
         tvaddhyAna-candrodaya-vardhamAnam
    amAntam Ananda-payodhimantaH
         payobhirakshNAM parivAhayanti 

[18.txt] 

    svairAnubhAvAH tvadadhIna-bhAvAH
         samR^iddha-vIryAH tvadanugraheNa
    vipashcito nAtha ! taranti mAyAM
         vaihArikIM mohana-pi~NchikAM te

[19.txt]

    prA~nnirmitAnAM tapasAM vipAkAH
         pratyagra-niHshreyasa-saMpado me
    samedhiShIraMstava pAdapadme
         sa~nkalpa-cintAmaNayaH praNAmAH 

[20.txt]

    vilupta-mUrdhanya-lipikramANAM
         surendra-cUDApada-lAlitAnAm
    tvada~nghri-rAjIva-rajaH-kaNAnAM
         bhUyAn prasAdo mayi nAtha! bhUyAt 

[21.txt]

    parisphurannUpura-citrabhAnu-
         prakAsha-nirdhUta-tamonuSha~ngAm
    padadvayIM te paricinmahentaH
         prabodha-rAjIva-vibhAta-sandhyAm 

[22.txt]

    tvatki~nkarAla~nkaraNocitAnAM
         tvayaiva kalpAntara-pAlitAnAm
    ma~Nju-praNAdaM maNi-nUpuraM te
         ma~NjUShikAM veda-giraM pratImaH 

[23.txt] 

    saMcintayAmi pratibhA-dashAsthAn
         saMdhukShayantaM samaya-pradIpAn
    vij~nAna-kalpadruma-pallavAbhaM
         vyAkhyAna-mudrA-madhuraM karaM te 

[24.txt]

    citte karomi sphuritAkShamAlaM
         savyetaraM nAtha ! karaM tvadIyam
    j~nAnAmR^itoda~Ncana-lampaTAnAM
         lIlAghaTIyantram ivAshritAnAm 

[25.txt] 

    prabodha-sindhoraruNaiH prakAshaiH
         pravALa-sa~nghAtamivodvahantam
    vibhAvaye deva ! sapustakaM te
         vAmaM karaM dakShiNam AshritAnAm 

[26.txt]

    tamAMsi bhittvA vishadairmayUkhaiH
         saMprINayantaM viduShashcakorAn
    nishAmaye tvAM navapuNDarIke 
         sharadghane candramiva sphurantam 

[27.txt] 

    dishantu me deva ! sadA tvadIyAH
         dayAtara~ngAnucarAH kaTAkShAH
    shrotreShu puMsAm amR^itaM kSharantIM
         sarasvatIM saMshrita-kAmadhenum 

[28.txt] 

    visheShavit-pAriShadeShu nAtha !
         vidagdha-goShThI-samarA~ngaNeShu
    jigIShato me kavitArkikendrAn 
         jihvAgra-siMhAsanam abhyupeyAH 

[29.txt] 

    tvAM cintayan tvanmayatAM prapannaH
         tvAm udgR^iNan shabdamayena dhAmnA
    svAmin ! samAjeShu samedhiShIya
         svacchanda-vAdAhava-baddhashUraH 

[30.txt] 

    nAnAvidhAnAm agatiH kalAnAM
         na cApi tIrtheShu kR^itAvatAraH
    dhruvaM tavAnAthaparigrahAyAH
         navaM navaM pAtram ahaM dayAyAH 

[31.txt] 

    akampanIyAnyapanIti-bhedaiH
         ala~nkR^iShIran hR^idayaM madIyam
    sha~nkAkaLa~nkApagamojjvalAni
         tattvAni samya~Nci tava prasAdAt 

[32.txt] 

    vyAkhyA-mudrAM kara-sarasijaiH pustakaM sh~nkha-cakre
         bibhradbhinna-sphaTika-rucire puNDarIke niShaNNaH
    amlAnashrIramR^itavishadairaMshubhiH plAvayan mAM
         AvirbhUyAdanagha-mahimA mAnase vAgadhIshaH 

[33.txt] 

    vAgartha-siddhi-hetoH   paThata hayagrIva-saMstutiM bhaktyA
    kavi-tArkika-kesariNA   ve~nkaTanAthena viracitAm etAm 

[Valedictory Tanian on Swami Desikan by Kumara Varadacharya]

    kavi-tArkika-simhAya        kalyANa-guNa-shAline
    shrImate ve~nkaTeshAya      vedAnta-gurave namaH