My kitchen sink overflow (you know the really small drain bit at the side of the sink) is full of really smelly yucky water. I have looked under the sink and it looks like a small pipe that I can take off (there is a screw going into the center of the overflow bit)
I have some caustic soda I was going to use but the problem is getting it into the overflow, thinking of using a small pippet or similar but then it would all just drain out and the water is up to the top. Also you have to use 1 litre of water and I would be there forever pippetting it into the overflow!
Does the main sink drain fine? If so then probably just the overflow blocked (toy car or similar object).
I'd put a large bucket under the sink trap positioned so it collected as much water as possible, when you disconnect the overflow pipe from where it connects to the sick trap/drain pipe.
Once the pipe is dangling you can poke something through it to dislodge whatever is stuck and then flush through with water. A small empty shampoo type bottle can be handy, as when filled with water and squeezed it produces a good jet of water.
Note: I wouldn't disconnect the pipe from what you describe as (the screw going into the center of the overflow bit). It's the other end that you need to disconnect, the bit that drains the water to the sinks main drain.
You may need to clean the trap under the sink. That's the blobby plastic thing! that has the u-bend inside. Connects to main drain on sink. Main drain is connected to overflow obviously. I poured bleach into ours and smell has gone. Apparently if you pour a cup of soda crystals and boiling water down main sink drain every week it keeps it clear.