Why is the Black Pearl partially unharmed at the begining of "At World's End"?

Keep in mind that Jack was in hell. You can argue that the locker is more like purgatory/limbo; but regardless of that discussion, Jack was clearly put in a situation that was meant to torment him.

Jack has the Pearl, but he's stuck in a desert and there's no wind. For someone like Jack, who lives to sail on the ocean, this is the worst that could happen. Eternity without sailing.
Before the gang show up (at least before Jack knows about it), the stone crabs had already put the Pearl back into the water. But there was still no wind, so the Pearl was still dead in the water.

Jack's torment seems intentional, not coincidental. That means that whoever created this environment (Davy Jones? A higher power?) would likely also have chosen to give Jack the Pearl to make it even more aggravating.

This means that the Pearl (in the locker) is not necessarily the actual Pearl, but rather a recreation of the Pearl (as Jack knows and loves it).

Note: it's also possible that the original Pearl was taken to the locker and then fixed by whoever designed Jack's torment; instead of creating a new Pearl from scratch.

And when the gang succeed in extracting Jack from the locker, they also succeed in extracting this recreated/fixed Pearl into the real world.

I think I know the answer. Let's look at it like this. When we see the kraken take down the pearl, did it crush our favourite ship? No, no it did not. The Black Pearl may be a powerful ship, with its speed and firepower, but it's not invincible. The Pearl was not repairs, nor destroyed, it was just pulled under the water. The Pearl we see has all the same damage to the cabin, stern lantern and other damage. It is the same ship. Of course, I could be entirely wrong, but my answer is likely to be accurate. So long story short: the Pearl was not destroyed, it was just pulled under

Preface: The following is a mostly speculative answer, but one that I would argue is logical and will try and support it with other POTC-related media. The problem lies in that POTC is an ongoing franchise, which may be working towards a redemption arc for Captain Jack Sparrow and includes each film digging deeper into Jack's past and ultimately is building some origin mythologies. So there could be a more solid answer down the line.

The Black Pearl was formerly The Wicked Wench and belonged to Captain Morgan. We also learn from DMTNT that Morgan was also the previous owner/user of Jack's MAGICAL compass. At some point, Morgan fights against Spanish Royal Armada and his ship ends up in Davy Jones Locker. Jack later makes a deal with Davy Jones, that includes a life-debt of a 100 souls (or his) for the ship to be risen and rechristened The Black Pearl. So we could assume a couple of things.

1. The ship previously had magical properties, as POTC-Wiki sites the Wicked Wench as a "legendary" ship and The Black Pearl as a "ghost" ship, therefor, it may be that Davy Jones (alone) could not ever fully destroy this ship, he can only take it out of commission. Also, why else would Jack want this ship, if it wasn't already special?
2. Davy Jones Locker is a magical space where most things are not destroyed, but only if he (or any Dutchman Captain?) is the one to put people and objects in there. The reason I say that is because clearly, the ship used by Barbossa, Elizabeth, and Will (AWE) does not make it, at least not that we know of (it's possible that Davy Jones Locker is vast and we have only seen a small part of it)--but that could have to do with the entry way, more than just that everyone who dies goes to the Locker, as we know most people are "suppose" to get ferried to some "other side". This is a responsibility of any Captain of the Flying Dutchman.

The Black Pearl was an infamous pirate ship, originally named the Wicked Wench. At one point it was sold and turned into a merchant vessel before turning back to piracy when it was re-christened. With sails as dark as a moonless night, and a hull painted to match, this legendary ship of the seven seas was every inch a pirate vessel. Built for action, this ghost ship could outsail any other sailing vessels...

The Wench's first captain was Morgan, a pirate who died in a battle with the Spanish Royal Navy. Command was later given to Jack Sparrow, who led the Wench and her crew on several voyages from West Africa to the Caribbean and back for the East India Trading Company on the behalf of Cutler Beckett. However, after Jack liberated a cargo of slaves, Beckett ordered the Wench set alight and sunk and her captain forever branded as a pirate. After failing to rescue the Wench, Jack made a deal with Davy Jones, the ghostly captain of the Flying Dutchman. Jones raised the Wicked Wench from the depths, giving Jack thirteen years as captain, in exchange for one hundred years of service aboard the Dutchman. http://pirates.wikia.com/wiki/Black_Pearl

At some point in his life Morgan became a captain of a pirate ship called the Wicked Wench. The Wicked Wench and her crew were part of the pirate fleet that tried to end Armando Salazar's reign of terror when the villainous Spanish capitán attempted to clean the seas from piracy. In a fierce battle off the coast of the mysterious Devil's Triangle the last pirates in the Caribbean joined forces in an attempt to sink Salazar's ship, the dreaded Silent Mary, but the mighty warship and her crew proved to be unstoppable.[3]

Soon, most of the pirate fleet was burning and sinking, and the Wicked Wench was the only pirate ship left afloat. Mortally wounded when the broadside from the Silent Mary hit the Wench's quarterdeck, Morgan gave his compass to Jack, an 18-year old boy, telling him never to betray it.[3] http://pirates.wikia.com/wiki/Morgan

Some Trivia that may support #1 - The Wicked Wench was already "magical."

In addition some trivia that might be of interest includes the canceled video game, a prequel, Pirates of the Caribbean: Armada of the Damned. The Game featured a new protagonist, Captain James Sterling who gets "cursed" by "Cursed Spanish Ghost Pirates" from a previous era. In the game, Sterling was to "reassemble" a magical wrecked and scattered Pirate Ship called, the Nemesis --AND depending on the choices one would make, Sterling would become a good-looking heroic legendary pirate or he would become feared and dreaded and turn into a ghostly-dark pirate who used a GIANT ANCHOR as a weapon!!

It is unknown who was the original captain of the Nemesis, but some time after his resurrection, captain Sterling has found the wreck of the Nemesis on some beach[2]. He decided to save the craft and turn it into his own pirate ship[3]. Pieces of the Nemesis were scattered all around the Caribbean, and Sterling spent many months in search of them.

A LOT of the game's elements seemed to be built back into the franchise, especially Dead Men Tell No Tales with Salazar in particular looking a lot like the "evil" version of Sterling and similar character to the games villain, Admiral Maldonado. The Black Pearl's anchor is featured also, as Carina Smith climbs up it!

A nameless pirate with an anchor on his back, member of the Drifting Dead, can be seen in Davy Jones' Locker in At World's End (video game).

In one of the other already-existing POTC video games, there is a GIANT ANCHOR seen in Davy Jones locker--some speculate there would of been a correlation between The Spanish cursed Pirates, Sterling, and Davy Jones, but no one will ever know since the game was never released, only beta-tested, but it's possible that they are taking these ideas and re-contextualizing them back into the newer parts of series with Morgan and his Magic Ship being a kind of equivalent with Sterling and his magic ship and that Jack's Sparrow's story relies on the history of Morgan and his magical objects.