I'm working on this question for a research paper, and I'm especially struggling to find sources that could give me insight into the specifics of how Rasputin negatively impacted the czar and czarina's decisions leading up to Nicholas II's abdication. I've found a lot of general stuff, i.e. that the influence of Rasputin was toxic to the czar's rule, but nothing specific. The best that I have right now is the book Russia and History's Turning Point by Alexander Kerensky, but even that is a little too general. Any help would be appreciated, especially if there are any good secondary sources.
Orlando Figes in his book A People's Tragedy contends that it was the erosion of public trust in the autocracy that was one of the pillars that broke during the Revolution. He posits that Rasputin was one of those erosions because he symbolized the corruption and wanton debased nature the autocracy.
To put it more plainly, Rasputin was a symbol of corruption.
He existed as this sort of propaganda bogeyman, that he was sleeping with the Empress and while also puppeteering the rule of the Romanov's. He was kind of this lightning rod of the ills of the Romanov's, especially since his entire reason for even being in the court was secret. Instead the outside saw this sexually depraved, pietist, heretical monk consorting in the court and potentially being a lover of the Empress while the Emperor was away. Compounded upon the fact that the Empress was already disliked by the majority of her subjects.
Now, this is not to say that Rasputin was the cause of the revolution. For we have already seen that revolution happened in his absence back in 1905. Instead he was more of a catalyst, another straw on the camels back. His importance is often romanticized because of how interesting of a character he was.