If I have K'rrik, Son of Yawgmoth on the battlefield and I am trying to cast a spell such as Bolas' Citadel, could I pay 3 of the black mana with 6 life, then cast a spell such as Culling the Weak to sacrifice K'rrik and use that mana to pay for the remainder of the Citadel? This doesn't make too much sense to me but someone mentioned to me it was possible. I know there are things such as LED which is exploitable because you pay costs after the spell is already on the stack. So is this possible, and if yes, why?
No, that is not possible. This is because you can't cast other spells in the middle of casting a spell so there is no opportunity to cast Culling the Weak in the middle of trying to cast Citadel. They might be thinking of using an ability with K'rrik and sacrificing K'rrik as part of it (say, using Phyrexian Tower after using K'rrik to pay for part of it). This also doesn't work since K'rrik doesn't apply until you start paying the costs but mana abilities, like Tower, need to be activated prior to that.
Also, I don't know what you think is exploitable about LED but it isn't. You can't use the mana from LED to cast a spell from your hand due to the "only any time you could cast an Instant" wording on the card which is there to expressly prevent that type of thing and, basically, turns that into this same situation you are asking about regarding K'rrik.
Also, I don't know what you think is exploitable about LED but it isn't. You can't use the mana from LED to cast a spell from your hand due to the "only any time you could cast an Instant" wording on the card which is there to expressly prevent that type of thing and, basically, turns that into this same situation you are asking about regarding K'rrik.