When a Devil Fruit user is locked with Kairoseki, is there any affect on his Haki abilities. Like in episode 627, couldn't luffy use Conqueror's Haki when he was in the cage made of Kairoseki?
Kairoseki has no effect on haki. Rayleigh could use Haki while being chained in the auction house ( during the sabaody archipelago arc).Also haki is inborn quality in every living thing in one piece world as said by Rayleigh. And kairoseki only affects devil fruit users. Non devil fruit users are not affected by kairoseki. So they can use haki however they like still being chained.
It wasn't really explained. We still don't know about the full extent of Haki or Kairōseki. Plus episode 627 is a filler so I wouldn't take what happens in there seriously, it is not canon. The anime is notorious for contradicting itself. Oda might explain Haki and Kairōseki more in the manga later on.
As i see it, the Kairoseki (sea stone was a translation i once memorized) makes the devil-fruit user physically weak, unable to move. And as you know, Luffy's ability is pretty much physical in any aspect, meaning when he feels weakened in his body, he also starts to feel helpless in his mind (until emotions tend to make him go berserk). Apparently that's just enough to make him lose his inner cool and not be able to use haki as he wished to.
Just my 2 cents...
According to Smoker, Kairoseki works to incapacitate devil fruit eaters because it gives off the same wavelength as the sea. In addition, Shanks used Haoshoku no Haki on the sea monster that took his arm while swimming. That means Haki is unaffected by the sea's wavelength and therefore unaffected by Kairoseki.
There are many situations where Luffy could use Haoshoku no Haki and chooses not to, especially on Whole Cake Island. So I think it is fair to assume he cannot use it in every situation. Because we cannot yet know exactly what situations he can use it in, we should look at the situations in which he does and compare. Everytime Luffy uses his Haoshoku no Haki, he is in one of two states: either in a totally desperate situation or in perfect health while calm and collected. In episode 627, however, he is not in a totally desperate situation (although Chopper is being beaten, he is not in danger of dying) and Luffy is not perfectly calm and collected.
It may be fair, then, to assume Luffy could not use his Haoshoku no Haki in this particular instance.