The second song inA song from my teenage years that I still remember clearly wais "Naval Port Night".

If you say “the second song” it implies you’re counting songs and have previously written about the first song. Since you haven’t and make no mention of more songs it’s better to just use a song.

During my middle school time, I didn't have any entertainment and I had to study hard, so I had little time to enjoy listening to music.

One day, I heard that one of my classmates was humming a song.

The song intrigued me so I was very curious about the songked him about it and he told me ithat the song was "The Naval Port Night".

I think it’s good to clarify that you asked him about the song since you were curious about it since otherwise there are no clear connections between the parts of the sentence separated by the and

Since then, I have tried learning to sing it in my spacre time and finally I could sing it, after some time, I could pull it off.

You have a little grammatical error. Either you “have” tried learn”ing” something or you tried “to” learn. I think you combined the two past tenses

Now, I want to translate ithe song’s lyrics into English to share withfor you.

I’d specify that you’re talking about the song instead of an “it” just out of stylistic reasons

The sea-waves gently shake the battleship gently.

The young sailors have sea waves under their heads while sleeping.sleep on the rough sea

HOh how hard the young sailors worked on the voyage.

HOh how hard the young sailors worked on the voyage.

Our warship is going to weighset anchor again.

The second song infrom my teenage years that I still remember clearly wais "Naval Port Night".

You're talking about remembering it now and talking about the song in the present tense, so I would use present tense

One day, I heard that one of my classmates was humming a song.

"I heard that..." It's an expression used for gossiping. Implying someone else told you about your classmate, not that you experienced it with your own ears. So replace it with just simply 'I heard'

I was very curious about the song and he told me that the song was "The Naval Port Night".

At first you introduced the song as 'Naval Port Night', now this time you included an article 'THE Naval Port Night', choose one definite title and use it throughout the text

He also told me that the song was very popular at that time.

You are almost quoting him, therefore it's in the present tense and should not include "at that time" because this is an expression used to talk about past events

Since then, I tried to learning to sing it in my spacre time, and then finally, I could sing it.

Remember - when we used a verb you must use an infinitive (basic) verb for the second one after the first. I tried (first verb) to learn (second, infinitive/basic)
You got it correct with 'to sing', so continue this on.
I would clarify also when you learned how to sing it. Examples: After some time, after a while, after a week/month/year, etc.

The sea- waves shake the battleship gently.

You do not need a hyphen here.
I would use a synonym for 'shake', such as 'rock', 'sway' or 'cradle' because you cannot really shake something gently

The sea waves shake them gently.

Same problem - I would change shake

The sea waves shake them gently.

^ Again

Until the morning sun shines on the sea

Until the morning? Sun usually shines from the morning onwards, I think this may be a stylistic error or incorrect translation

Our warship is going to weigh anchor again.

Not sure what you meant to say here, my apologies.