There is no doubt as to the healing efficacy of the 'ApadAm' mantra. My doubt is only as to its meaning as given by Thomas Ashley-Farrand.
Firstly, the mantra is found as the 35th verse of Sri Rama Raksha Stotram, a Hindu-Sanskrit prayer composed by the saint Budha Koushika, who quotes from other great stotras. (http://www.hindupedia.com/en/Sri_Rama_Raksha_Stotram)
I am not able to find an audio file of the mantra as Thomas Ashley-Farrand would have it chanted, but assuming that the correct intonations are maintained, the following could be the phrase and word match of the version as he has given:Om Apadamapa Hataram Dataram Sarva Sampadam
Loka Bhi Ramam Sri Rama Bhuyo Bhuyo Namamyaham
The very rough translation is, 'Om, Oh most compassionate Rama please send your healing energy right here to the earth, to the earth (twice for emphasis.)'
Apadaam - no match in the given meaning;
this is actually the sandhi extension of term 'Apat' which means 'hasten towards, rush in, assail, fall out, happen' (thus indicating an accident, emergency, danger or allfliction).
(apa)Haataram - 'healing energy'? The term 'apahata' only means 'destroyed, warded off' (MWD). The term 'hAta' means 'given up, abandoned'. If it is 'hata', it would mean 'struck by lightning, injured, beaten, visited or afflicted, tormented by, (sexually) violated, ruined, undone, worthless, defective, deprive of, killing, hurting...'
The actual phrase of the mantra is 'apahartAram'--not 'apa Hataram'. 'apa' is the prefix meaning 'away, off, back' and 'harta' means to 'destroy, seize, take'.
Some versions give the term 'hantAram', from the verb 'hant' meaning 'destroy, strike down, kill'. The Sanskrit text of the Rama Raksha Stotra, however, has only the word 'apahartAram' (http://sanskritdocuments.org/all_pdf/rraksha.pdf)
daataram - give, 'send', bestow (this seems to be right)
sarva sampadam - this phrase could match with 'healing energy' though it literally means 'all the wealth', because the very word 'wealth' derives from 'weal' (well-being, welfare).
loka bhi ramam sri rama - 'Oh most compassionate Rama'? The term 'loka' means 'world, people'; 'abhirAma' means 'pleasing, beautiful, agreeable' which is not the same as 'compassionate'. 'shrI rAmam' indicates that Rama who is Vishnu, is always accompanied by his consort Sita, who is Lakshmi.
bhuyo bhuyo - 'bhu' may mean 'earth, becoming, being, arising, springing' but the phrase intended here by the author of the stotra Budha Koushika is only 'bhUyo bhUyo'. Now, the term 'bhUyo' is derived from 'bhUyas', which means here 'again, once more, still more, first-then-next, abound, becoming in a greater degree'.
namamyaham - salutations.
Thus, fhe following points emerge from the foregoing:
• Sri Rama, of course, is extremely compassionate but this meaning does not figure in the mantra.
• A prayer for healing is usually personal, chanted either by the afflicted or by an agent. A personal prayer usually does not ask the deity prayed to 'send your healing energy right here to the Earth'; instead, it supplicates for personal healing.
• Thomas Ashley-Farrand might have an impressive spiritual record and perhaps be a mystic, but his Website (http://www.sanskritmantra.com/)is commercial (there is nothing wrong with it, though I would prefer much more 'freebies' as in the Website of Divine Life Society or that of Swami Krishnananda). His audio works and mantra dispensations for healing might have very good effects, but he cannot and should not have "very rough translation" of the mantras he dispenses, tailoring the meaning to suit his needs.
• This brings us to the valid point MahaHrada has made repeatedly: proper chanting of the mantras cannot be had from books and the Internet, but best from a guru. Where a guru is not in sight, mantras should be had from authentic audio chantings given by Hindu scholars. I invariably find the audio chants of Western scholars lacking in clarity and depth, because they pronounce Sanskrit words in the way the English words are spoken, usually with undesirable slurs. If a Brahmin pupil in a Vedic Pathashala does not pronounce a Sanskrit word with the correct sound and intonation, the teacher usually chides him with the words, 'Burn your tongue with dharba grass, only then it will be rid of the slurs!" Such is the clarity required of Sanskrit pronunciation.
If sound and intonation is not important in mantra, one can as well pray with just its meaning.
I am not against the efforts of any Western Hindu gurus. I only insist that they keep up the Hindu traditional forms, expressions and meanings in whatever they seek to do.