This is one of those things that's going to be up to the GM, and very much dependent on whether you're running a more pulp or more purist game. By default, the gods are all somewhere on a spectrum between rational independent entities and innate cosmic forces. And killing a cosmic force (even the personified expression of a cosmic force) might be a bad idea to begin with. However, engaging a god in its home (where it isn't manifested and has full power,) or finding a way to bind it, or pulling it into a fight with another god could all work.
In purist cosmic horror, winning against the gods---even in the short term---is generally treated as impossible. I tend to lean more pulp, with the gods as impossibly powerful and uncaring monsters, but still capable of being outplayed sometimes. In cosmic horror stuff I've run, players *have* worked out ways to kill gods---usually at great cost, and with immense risk.
Again, this is ultimately going to be up to the GM. For me personally, if the players come up with a means of attack that I can't imagine the god has sufficient defenses against, and they deliver it in a way that the god can't distance itself from, it'll work. And if a god falls, its Miseries are repealed---although they might be replaced with other ramifications from killing it.