newish to raids, how does conditional finality drop work?

[maiden obsidian]

Can I get another shot at it if I kill nezzy again? Killed him last night for a first completion on my hunter.

[round dune]

u can only have 1 try per character every week u can do the triumphs for the seal to increase the drop chance of the exotic idk how much the boost is but i was doing them and got it on my 9th run