Have you tried using the homemade drain cleaner solution of baking soda, hot water and vinegar?

Dump half a box of baking soda in the drain, then pour in a quart bottle of any kind of vinegar (I like clear). It will fizzle a lot, when it settles run lots and lots of hot water.I got this tip from my plumber. He said doing this once a week will maintain your sink. I do it probably once in 2 months.

Thanks, that works for the main pipe/drain, but this is around the sides. Will have to find a way to get the baking powder and vinegar in there, I guess.

Inquire at a good hardware store where there are helpful clerks. Ask for a small circular bronze or brass tube brush. You might find what you need in the sporting good section as a gun cleaning brush.

Here is a link that might be useful: Long Brass Tube Brush

I think you are describing the four outlets into the basin drainpipe from the sink's overflow system.
These drain holes tend to catch debris and sometimes build up slime mold over time.
Cleaning them is a three step process.

First, you try to determine if you have enough flow by filling the sink and allowing the overflow outlet to cope with your normal tap settings. If OK then drain sink:
The next step involves plugging the sink drain and the small overflow drains with something like a pill bottle wrapped in a plastic shopping bag to stop all draining. Then fill up the overflow system with a sanitizing bleach solution and leave for 10 minutes or so. You might want to have some water in the basin to avoid bleach staining from any spillage.
Then pull the pill bottle and let everything drain.
If any loose clumps show up just pick away at them gradually over time with a small screwdriver.
Don't get aggressive with anything that might be a sealant.
Soap scum and slime mold are greasy and other stuff tends to resemble wet cardboard so doing it every week or two for a few months plus sending hot water down the over flow from a full basin should take care of the problem.