I recently realized that two of my toilets have very weak flushes due to the clogged holes around the rim. I tried unclogging them with a paper clip or a wire hanger to no avail. I'm pretty sure it's not the holes themselves but what inside the rim (perhaps some mineral build-up---the house is over 20 years old) that's preventing the water from draining. I could visually see that the water flows out of the other holes on the right side but no water flow apparent on the left.

Is there something I could buy to remove or reduce some of this build-up that perhaps I could let sit in the tank?

Turn the water off to the toilet and empty the tank. Lift the flapper and pour a 1/2 Gallon of bleach down the hole and let it set overnight. Next morning take a coathanger and clear the rim holes. Have you checked the jet? Look down at the bottom of the bowl. If there is a small hole, then that is a jet that starts the syphon action. If it's clogged the water will just swirl around and slowly go down leaving solids behind. Take your finger,(UGH! ) and run it around the inside of the opening. Over the years minerals build up and cut down on the syphon (flush) action. If it is rough or you feel build up, take a table knife and put a bend in it to get around the curve in the bowl and chip and scrap it clear. And last, check the water level in the tank. It should be 1/2" below the top of the over flow tube. Good luck, Tom