Clarke conceded a late penalty against Brentford on Tuesday night, which led to the comments, and Waggott has backed his player's comments that the abuse was unacceptable.
"I can take criticism when it's about football," Clarke told Coventry's official website. "However there is just no place in football or society for racism, it's as simple as that."
Waggott was equally damning of the tweets. "I think most players accept that if they engage in social media there are both positives and negatives," he said.
"While criticising an individual player's performance in a match is not particularly helpful or constructive, we cannot stop that.
"However, in these instances, they went way past the mark and were no longer criticism of a player or performance but out-and-out abuse and I am sure every right-minded Coventry City supporter would condemn such behaviour."
City are currently struggling in League One and sit 21st.