My goal is to post roughly 20 lines of the poem each post. I'm only going to do the 訓読 based on Hagiwara's citation of the poem; this uses some different kanji and a few different kundoku from the 新釈漢文大系 (SKT) text that I'm using for annotations. If you want to see the original Chinese text and pinyin, go to the Wikisource page. That page has a good natural translation so I'll try to be a bit more literal and explain some more of the terms and such.
長恨歌 (Song of Everlasting Regret/Neverending Sorrow)
The King of Han valued sex, and wanted a beautiful woman.
- The "King of Han" is a poetic conceit to avoid naming Emperor Xuanzong, since the Tang Dynasty was still ruling. There is no doubt who the poem is about, though. The suggestion is that this is about Emperor Wu of Han and his relationship with Consort Li. Li has been suggested as one model for the Kiritsubo Intimate. The term 傾国 also comes from the Book of Han's biography of Consort Li; see the translated passage at wiktionary.
- "sex" is kind of a bold translation of 色 but I couldn't come up with a better one. "Love" or "romance" don't really work. Maybe the wikisource's choice of "dalliances", although somewhat Victorian, is more appropriate.
He searched during the many years of his reign, but couldn't find any such beauty.
In the House of Yang there was a daughter who had just come of age.
She was raised and kept in the inner house, and people did not yet know her.
Her natural beauty was difficult to ignore.
- I'm not entirely sure how to interpret 自難棄; apparently it's just "impossible to ignore" but I don't fully understand the construction of that meaning.
One day, she was chosen, and went by the King's side.
By turning her head and laughing once, it was one hundred times as sexy.
- None of the editions I've looked at really translate the meaning of the words, they just go with something like "Her entrancing smile could melt anyone's heart". I'm really not sure if my more literal interpretation is accurate at all.
The powdered beauties of the palace had no beauty.
- Presumably in comparison to Yang Guifei. A less literal translation would be "The other women in the palace could not compare to her."
In the cold spring, the King gave her the gift of bathing in the Huaqing Pool.
The waters of the hot spring were smooth, and cleansed her white skin.
- 凝脂 is kind of an odd phrase; it seems to mean "(skin) white like solid fat" and is derived from the Shi jing.
Her serving women helped her exit; she was slender and had no strength.
This is just when she began to receive the King's favor.
-SKT reads 始 as まさしく and says it means "Just now".
She had hair like a cloud, a face like a flower, and gold jewelry in her hair.
- 雲 may just mean "dark" but it might also refer to other aspects of clouds as well.
They spent the spring nights behind a warm screen of hibiscus.
They lamented the shortness of the spring nights, and rose when the sun was high.
16 此れ従、 君王、早朝（あさまつりごと）せず。
From this point, the King no longer did his morning government duties.
-従 is evidently read より here.
She pleased the Emperor and accompanied him to banquets; she had no free time.
- SKT emends 宴 to 寝 from other manuscripts and says this means "she spent time in his sleeping chamber". 承歓 is evidently some kind of idiom.
In the spring she accompanied him on his spring outings, and at night she was with him every night.
There were three thousand beautiful women in the palace,
- SKT emends 後 to 漢 from other manuscripts, changing the meaning to "In the Han Palace..."
(but) the love for the three thousand was with one woman alone.
Sounds like a great love story! What could go wrong?