The "Shri Rama Mangalam" is attributed to Shri Manavala Mamunigal.
It does not appear in the often quoted set of his works, though.
It is composed of 12 shlokas in the anuShTup meter.

All 12 verses are benedictions to Lord Rama, and wish all auspiciousness to Him.
Lord Rama is referred to in the "dative case" for all benedictory verses.
The verses are set up to match the Valmiki Ramayana in a chronological fashion.

[1] The Practical Sanskrit-English Dictionary (Revised Enlarged Edition)
      by Vaman Shivram Apte, Motilal Banarsidass Publishers
[2] Source at prapatti.com
[3] Audio Rendition

Notes:
    - "ma~Ngalam" is translated as "Salutations" here; It has many meanings
    including "benediction", "auspiciousness", "blessing", "felicitation", etc.
    - When ma~Ngalam is used, the "dative case" is used for the object which
    is Lord Rama in this composition; so Lord Rama is kosalendraH and so
    "ma~Ngalam kosalendrAya"; ..mUrtiH -> ..mUrtaye; ..vihArI -> ..vihAriNe
    - It is not "rAj~Ne" anywhere since the "rAjan" shabdaH, on any
    samAsa-operation, will become ..rAjaH (like in mahArAjaH) and hence
    declines as "..rAjAya" in the dative case
    - The Sanskrit words are mentioned alongside the english translation 
    within [..] and unmentioned but intended words are placed within (..)

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    ma~NgalaM kosalendrAya mahanIya-guNAbdhaye
    cakravarti-tanUjAya sArvabhaumAya ma~Ngalam             (1)    
        

    Salutations [ma~Ngalam] be to
    * the leader [indraH] of the land of Kosala
    * the One who is an ocean [abdhiH] of great [mahanIya] virtues [guNa]
    * the One who is the son [tanUjaH] of the Emperor (Dasharatha) [cakravartiH]
    * the Great King who rules over all lands 
      [sarva-bhUmInAm adhipatiH - sArvabhaumaH]

======
    veda-vedAnta-vedyAya megha-shyAmaLa-mUrtaye
    puMsAM mohana-rUpAya puNya-shlokAya ma~Ngalam           (2)    
        

    Salutations [ma~Ngalam] be to
    * the One who is worthy [vedyaH] to be known by the Vedas and Upanishads
    * the One whose embodiment [mUrtiH] is like the black-hued [shyAmaLa] 
      clouds [meghaH]
    * the One whose enrapturing [mohana] form [rUpam] even attracts men [puMsAm]
    * the One who is auspicious [puNya] for praising/worshipping [shloka]
    
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    vishvAmitrAntara~NgAya mithilAnagarI-pateH
    bhAgyAnAM paripAkAya bhavya-rUpAya ma~Ngalam            (3)    
        

    Salutations [ma~Ngalam] be to
    * the One who is most beloved [antara~NgaH](disciple) of Sage Vishwamitra
    * the One who was the result/consummation [paripAkaH] of all the 
      prosperity/happiness [bhAgyam] of the King of the land of Mithila 
      [mithilA-nagarI-patiH] (Janaka)
    * the One who has an excellent, handsome [bhavya] form/features [rUpam]

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    pitR^i-bhaktAya satataM bhrAtR^ibhiH saha sItayA
    nanditAkhila-lokAya rAma-bhadrAya ma~Ngalam             (4)    
        

    Salutations [ma~Ngalam] be to
    * the One who was always [satatam] devoted [bhakta] to (serving His) 
      father [pitR^i]
    * the One who always [satatam] delighted [nandita] the entire world 
      [akhila-loka]along with His brothers [bhrAtaraH] and (wife) Sita        
    * the beloved praiseworthy [bhadra] Lord Rama

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    tyakta-sAketa-vAsAya citrakUTa-vihAriNe
    sevyAya sarva-yaminAM dhIrodArAya ma~Ngalam             (5)    
        

    Salutations [ma~Ngalam] be to
    * the One who gave up [tyakta] the (luxurious) living [vAsaH] in 
      Ayodhya [sAketa](nagaram) 
    * the One who enjoyed [vihArI] (living/wandering) in Chitrakuta 
      (literally means hills with various wonders)
    * the One who was served/worshipped [sevyaH] by all sages [yamI]
    * the One who is the perfect, illustrious [udAra] emblem of courage 
      and steadfastness [dhIra]
    
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    saumitriNA ca jAnakyA cApa-bANAsi-dhAriNe
    saMsevyAya sadA bhaktyA svAmine mama ma~Ngalam          (6)    
    

    Salutations [ma~Ngalam] be to
    * the One who wielded [dhArI] the bow [cApaH], arrow [bANaH] and sword 
      [asiH] (and wandered in the forest) with Sita [jAnakI] and Lakshmana 
      [saumitrI]
    * the my [mama] Lord [svAmI] who is the fittest to be worshipped [sevyaH] 
      always [sadA] with devotion [bhaktiH]     

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    daNDakAraNya-vAsAya khaNDitAmara-shatrave 
    gR^idhra-rAjAya bhaktAya muktidAyAstu ma~Ngalam         (7)    
        

    Salutations [ma~Ngalam] be [astu] to
    * the One who lived [vAsaH] in the Dandaka forest [araNyam]
    * destroyed [khandita] Devas' [amarAH] enemies [shatravaH]
    * the Granter [dAyaH] of Liberation [muktiH] to His devotee [bhaktaH],
      the King [rAjA] among Vultures [gR^idhraH] (Jatayu)

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    sAdaraM shabarI-datta-phala-mUlAbhilAShiNe
    saulabhya-paripUrNAya sattvodriktAya ma~Ngalam          (8)     
        

    Salutations [ma~Ngalam] be [astu] to
    * the One who expressed desire [abhilAShI] with respect and 
      eagerness [sAdaram] for the root [mUlam] of the fruits [phalam] 
      offered [datta] by Shabari
    * the One who is filled [paripUrNaH] with the quality of being easily 
      reached by others [saulabhyam]
    * the One is pre-eminent in courage and has overwhelming [udrikta] 
      goodness [sattvam] 

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    hanumat-samavetAya harIshAbhIShTa-dAyine
    vAli-pramathanAyAstu mahAdhIrAya ma~Ngalam              (9)    
        

    Salutations [ma~Ngalam] be [astu] to
    * the One who became intimately connected, united [samavetaH] with Hanuman
    * the One who granted [dAyI] the wishes [abhIShTa] of the King [Isha] of 
      monkeys [hariH] (Sugriva)
    * the One who tormented and vanquished [pramathanaH] Vali
    * the One who is the most valorous [mahAdhIraH] 
      
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    shrImate raghuvIrAya setUlla~Nghita-sindhave
    jita-rAkShasa-rAjAya raNa-dhIrAya ma~Ngalam             (10)    
        

    Salutations [ma~Ngalam] be to
    * the Illustrious Lord [shrImAn] who is the hero [vIraH] in the 
      lineage of Raghu
    * the One who (constructed) a bridge [setuH] to cross 
      over [ulla~Nghita] the ocean [sindhuH] (to reach Lanka)
    * the One who won over [jita] the King [rAjA] of Demons [rakShasaH] (Ravana)
    * the One who was the hero [dhIraH] of the war [raNaH]
        
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    AsAdya nagarIM divyAm abhiShiktAya sItayA
    rAjAdhirAja-rAjAya rAma-bhadrAya ma~Ngalam              (11)    
        

    Salutations [ma~Ngalam] be to
    * the One who attained/reached [AsAdya] the divine land [divyAM nagarIm]
      (Ayodhya) and became coronated [abhiShiktaH] (as the King) with Sita    
    * the One who is the King of all Emperors [adhirAjaH] and Kings
    * the Illustrious Lord Rama
    
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    ma~NgalAshAsanaparairmadAcArya-purogamaiH 
    sarvaishca pUrvairAcAryaiH satkR^itAyAstu ma~Ngalam     (12)    
    

    Salutations [ma~Ngalam] and blessings [AshAsanam] be to
    * the One who was worshipped, adored and respected [satkR^ita] by all
      preceding Acharyas [sarvaiH pUrvaiH AcAryaiH ca] with my Acharya 
      [madAcArya] at the forefront [purogamaiH];
      They deemed wishing the Lord well [ma~NgalAshAsana] as their highest 
      aim [paraiH]