There are a few reasons/solutions. on most cars there is a "cable" that goes from the pedal to the engine. If the cable gets corroded or dirty it will be clunky like you describe. Also there is a pivot point on the pedal, if there is dirt around the pedal it will stick. The only other possibility would the the butterfly valve you are operating (whether carb or throttle body) is not operating smoothly.

A great way to check would be to disconnect the cable from the pedal, then push the pedal with your hand and see if it moves up and down smoothly, if not, the problem is the pedal. then remove the cable from the butterfly and operate the butterfly. same as the pedal, if it moves smoothly, the problem is not there. Now that you have the cable disconnected from the pedal and butterfly, push and pull the cable through the outer jacket. Does it move smoothly?

I know that his sounds pretty obvious, but I've missed some pretty obvious things. First, check that the floor mat hasn't shifted in a way that binds the pedal, then proceed to the other suggestions made.

Head down into the footwell, clean and sparingly lubricate the pedal pivot. Lubricate all linkage pivot points, lubricate cable wherever you can. If that doesn't help, it's probably the cable which is worn and needs replacing.

In addition to the cables that others have mentioned, the throttle plate can get buildup around it. Give that a dose of your favorite spray cleaner.

My accelerator has a bit of sticking. Not much but it's enough so that when the stick slips, it jerks the car forward because it depressed more than I wanted. Is there a way to smooth this out?