Now it has becomen decades ago, but around that time there was a pair of birds coming almost everyday ato the kitchen window and reciting an interesting song/tune.

They were very modest in appearance and slim middle sized birdsslim, middle-sized birds, very modest in appearance.

Once, when I went outside to the ?small gallery at the light well? of the apartment building, I had a chance to see them very short distancup close.

I'm sorry - wasn't sure what you meant by small gallery at the light well. I'd be happy to offer a correction if you could clarify for me. :)

They were sitting on the fence there showingwith their back to me, and I witnessednoticed/saw their long tails.

There wereIt was several seconds before they flew away sayinging kakakakakaka...

I think they delayedit took them a little to notice me.

Their name of those birds turned out to be 'mourning birddove.'

ItThis was a very unexpected name to me because their voice was, soft but rather, eccentric and amusing.

They say hoot, hoot, hoot, like owls, but in a higher tone, then once in the phrase it. One of the hoots becomes about one octave higher and sayounds more like 'He'.

Now it has become decades ago, but around that timeDecades ago, there was a pair of birds coming almost everyday at the kitchen window and, reciting an interesting song.

"Now it has become decades ago, but around that time" sounds very unnatural, "decades ago" evokes the feeling of it being a long time ago without using so many words

They were very modest in appearance and slim middle sized birds, modest in appearance.

Not necessarily incorrect but this sounds more "novel-y"

Once, when I went outside to the small gallery at the light well of the apartment building, I had a chance to see them very short distanceclosely.

There were sSeveral seconds passed before they flew away saying kakakakakaka...

more natural

I think they delayfailed to notice me.

"delayed" sounds odd, doesn't really make sense. "failed" implies they didn't notice you which is what i imagine you were going for

It was a very unexpected name to me because their voice was, soft but rather, eccentric and amusing.

They say hoot, hoot, hoot, like owls, but in a higher tone, then once in the phrase it becomes about one octave higher and saythey go 'He'.

Now iIt hwas become decades ago, but around that time there was a pair of birds coming almost every day ato the kitchen window and reciting an interesting song.

every day (two words)

They were very modest in appearance and slim, middle sized birds.

Once, when I went outside to the small gallery at the light well of the apartment building, I had a chance to see them [from a very short distance | up close].

They were sitting on the fence there showing their backs to me and I witnesssaw / spied / discerned their long tails.

There were sSeveral seconds passed before they flew away saying kakakakakaka...

I think they delayed to notice me.

I don’t understand this. “They delayed” implies that they did this deliberately—they chose to delay. Is this what you mean?

Their name turned out to be 'mourning bird.'

Do you mean a mourning dove? Probably not, as they’re not at all slim.

It was a very unexpected name to me because their voice was,calls / songs were soft but rather, eccentric and amusing.

They say hoot, hoot, hoot, like owls, but in a higher tone, then once in the phrase it becomes about one octave higher and they say 'He'.

This description does sound like the call of a mourning dove. I learned to imitate a mourning dove’s song when I was younger and can still do it now. Sometimes they will answer me as if we’re talking. I have no idea what we’re saying, though. :-)