Thermal paste won't do anything except possibly make the temps of your CPU lower. POSSIBLY. If the initial application of thermal paste is good then it won't do anything to the temps.
Also, I said to check you temps when under load. If they are too high then it's possible thermal paste could lower the temps.
It is also possible that the stuttering is due to temps being too high. But again, you should be checking on that. Also, even with a proper application of some good thermal paste, if you are using a stock Intel cooler it won't do you any good since the cooler itself is garbage.
But I'll say it one more time so it's clear, changing thermal paste as if it somehow fixes issues in games makes no sense. Thermal paste only effects CPU temps and only if the initial application of paste is bad and it's not your cooler that sucks.
The only possible way doing this could remove issues like stuttering in games, is if your cooler is good but the paste is bad and your CPU temps are getting too high and it is causing thermal throttling on the CPU at which point a fresh and proper application of paste will fix the temp issues and thusly fix the stuttering, but ONLY if you have a good cooler and ONLY if the current paste is bad and ONLY if it's thermal throttling and THIS is what's causing stutters.
Otherwise, changing out the paste will do nothing for you. Absolutely nothing.