Could you correct the German-to-English translation by "Google translate"?

Mutter, hast du mir vergeben
Mutter, denkst du noch daran
Mutter, hast du mir vergeben, was ich dir angetan
Heimat, hast du mir vergeben
Heimat, denkst du noch daran
Heimat, hast du mir vergeben, was ich dir angetan
Das Glück lockte mich fort von hier
Fort von Heimat und Haus
Ich ging mit all den andern und kam nie mehr nach Haus
Mutter, kannst du mich noch lieben
Mutter, gib mir deine Hand
Bin dein Kind doch geblieben, fremd in fremdem Land
Mutter, ich will in die Heimat
Mutter, die Welt ist so groß
Mutter, ich will in die Heimat, nimm mich in deinen Schoß

Mom, you have forgiven me
Mother, do you still think it
Mom, you have forgiven me for what I done to you
Home, you have forgiven me
Home, you still thinking of something
Home, you forgave me for what I done to you
Happiness lured me away from here
Away from home and home
I went with all the others and never came home
Mother, can you still love me
Mother, give me your hand
'm Your child, but remained a stranger in a strange land
Mother, I want to go home
Mother, the world is so big
Mother, I want to go home, take me in your lap

http://www.lyricstime.com/marlene-dietrich-mutter-lyrics.html

Is that Heine? Not sure whether I’ve struck the right note, but here is a quick correction: (by the way, I take it there’s no copyright issue)

Mother, have you forgiven me
Mother, do you still remember
Mother, have you forgiven me for what I did to you
My country, (Fatherland, Motherland), have you have forgiven me
My country, do you still remember
My country, have you forgiven me for what I did to you
Fortune lured me away from here
Away from home and country (I’ve reversed the order here)
I went with all the others and never returned home
Mother, can you still love me
Mother, give me your hand
I am still your child, (remained) a stranger in a strange land
Mother, I want to come home
Mother, the world is so vast
Mother, I want to come home, take me in your lap

Thank you very much.

Can “Heimat” mean a place where a person was born that is not a country but an area in a country?

The song is one of Marlene Dietrich’s , which I listen to while I am driving.

Yes, Heimat is where you have your roots. Meine Heimat is in northern Germany, my home is in the UK. These days I feel very much at home in the UK, but die Heimat is calling from time to time and then I get Heimweh.

In the Japanese language, “ふるさと(故郷)” means “Heimat,” and ''ホームシック" means “Heimweh.”
“ホームシック” is one of “Katakana-English” words. Thank you for replying to my question.

Errata:
“one of ‘Katakana-English’ words” should read “one of the ‘Katakana-English’ words.”

According to the dictionary, 故郷 has two pronunciations: ふるさと and こきょう. Can these two pronunciations be used interchangeably or is there a difference in meaning?

Strictly speaking, ”故郷” should be pronounced “こきょう.” ”ふるさと” can be written “古里.”

I see. Thank you Yutaka-san.

There is hardly any difference in meaning between “ふるさと” and “こきょう.”

I have always felt that the Chinese pronunciation of words, i.e こきょう is more formal and the original Japanese pronunciation is warmer. I think of the line in 北国の春。。。。。 あの故郷にかえろうかな

See video and enjoy http://bit.ly/5ylBRZ

That`s right. “ふるさと” is warmer.