This happens consistently: I wake my iMac from sleep, see very high CPU usage, look in Activity Monitor, and notice that Web Clipper appears to be the culprit. I haven't confirmed this yet, but I *think* that some time after waking CPU usage goes back down to normal. I haven't confirmed it because I generally just find it easier to disable Web Clipper. (I keep Web Clipper disabled on my laptop for this very reason.)
As a guess, it seems to me that Web Clipper is trying to access the network during sleep, can't, and ends up polling for a connection. In doing some quick research I discovered the "Stop plug-ins to save power" setting which is apparently on by default. Related? I don't know; as I said, it's just a guess. (Of course, Web Clipper is an extension rather than a plug-in, but I don't know if Apple is being precise in its description of the setting.)
I'm using Web Clipper 7.12.3, Safari 13.0.5, macOS 10.15.3.