Can I use a toilet auger to clean my kitchen drain from the clean out?

I would not! It can get stuck in the kitchen pipes and would not be able to make any turns. The toilet drain pipe is 4". It needs something substantial to rod ir through. The kitchen drain pipe is 2". The proper size snake for a kitchen drain would be 1/4" to 1/2" flat or round.

Try to clean the drain clog with baking soda, followed by vinegar, let it soak for a while and follow by at least a gallon of boiling water. I tried this to no avail once, but ended up having the clog so deep down that I had a pro due it all the way. I have been using the baking soda, vinegar and boiling water since and have not had another problem. We have the drain pipe for the kitchen sink running down the wall in the basement and it has a cover you can take off and see inside the pipe, It is spotless, without any kind of goop or nasty stuff growing on it. The pro was very surprised to see it that clean when he looked to make sure the snake was going down the pipe and not outside(the outside faucet is right below the sink). He told me to keep doing this occasionally to keep the pipes clean and maybe nothing will clog it again. The clog was almost to where the pipe went outside to connect to the sewer.