Merhaba! I imagine I have nothing correct in this song, but I´ve made a try anyway. I see in the dictionary that "nazlı" can be "one who feigns reluctance." That would seem to work here, but I don´t know how to express that in one word in English. I also know that "ay" can mean "moon," and I have seen translations on the web that use "moon," but that doesn´t seem quite right to me. ... But I´m sure you will tell me. ... Teşekkürler. 

Nazlı Ay -- Oh, Coquette

Ay nazlı ay -- Oh, coquette, oh
gelin ay -- Oh, bride
Git ona söyle -- Go, tell him
Ah gücüme gidiyor yalnızlığım böyle -- Ah, in this way, my loneliness will be my strength
Ah bu inadından az mı ağladım ben -- Ah, have I cried a little because of your obstinancy?

Ay ay nazlı ay -- Oh, oh, coquette, oh
Git onun ülkesine -- Go to his country
Git onun beldesine -- Go to his city
Ay ay git söyle -- Oh, oh, go, tell him

http://www.youtube.com/watch?v=vlxTQVTHeuI

Ay can mean oh and moon as you wrote. Maybe he uses both meanings in the song. I am not sure about that.

Thank you, gokuyum. Translating songs is full of trouble for me. So, I have questions for you:

"Git ona söyle -- Go, tell him/her"

How do you know that it is "her" and not "him"? I am guessing -- obviously -- but this song sounds like a jilted lover telling the girl to go her way, to go to the man she has chosen over the singer. 

"Ah gücüme gidiyor yalnızlığım böyle -- Ah, in this way, my loneliness offends me ( look at gücüne gitmek)"

I cannot find this phrase in my dictionaries. But I do find "gücendirmek" and "gücenme." İmdat!

"Ah bu inadından az mı ağladım ben -- Ah, have I cried a little because of your obstinancy? ( this sentence means I have cried a lot )"

I do not understand this. "Az" means "little," yes? And "mı" is the question particle, yes? So, is this a colloquialism that means the opposite of what it looks like? "I cried not a little because of your obstinancy." Anlat, lütfen.

And one last question: Is Turkish as filled with ambiguities for Turkish speakers as it is for English speakers? Of course, poetry nearly always has ambiguities -- it is supposed to make you think. But this question of "ay" seems very open-ended. For instance, does "moon bride" have a special meaning in Turkish?

Thank you for putting up with all my questions.

Bu sözlük için çok teşekkür ederim, gokuyum. Ona çalışacağım. 

Turkish songs can be mystifying to me, but I still love them. Mabel Matiz, in particular, seems to use lots of very colloquial phrases, so I can´t always understand his songs. But he is a wonderful musician. He is my current favorite.

Speaking of wordplay in songs, here is what I think must be a Turkish pun, from a song that Barış Manço sang early in his career. The song is "Fil ile Kurbağa." An elephant falls in love with a frog, but their romance ends in disaster. The last line of the song is:

"Koskoca aşkı vücudu gibi çok ağır geldi."

I think this means: "A very great love, like his body, was too difficult." The dictionary says that "ağır gelmek" can mean "to be too difficult." But the pun is in the word "ağır," yes? Heavy, man.

http://alkislarlayasiyorum.com/icerik/8813/baris-manco-fil-ile-kurbaga-1972