Check the task manager when your playing games. How many % of the RAM is being used? If it is on Max, then yes it will make it faster and smoother, and even better FPS.
If not, then you will need to upgrade your Graphics Card, in which case you will also probably need to upgrade the RAM as you will be running games at higher resolution and graphic settings and 4Gb is not enough.
Nvidia 820m is a laptop GPU and you can't change a laptop graphics card. Also upgrading to 8gb ram may improve some stutters but it wont give you smooth gameplay since your current GPU is very weak. Older games will run fine though.