I'm not sure what your threads are supposed to do with the currently running requests. I would let them finish their current request and have them recheck a global, shared flag whether to pause or (re)start processing. A more convoluted alternative would be to $queue->unshift dummy jobs on the threads work queue that tell each thread to pause.
If the request really is "stop whatever you're doing", have a look at the "suspend" example in threads. It seems to use signals, but as the signals are claimed to stay within the confines of Perl and seem not to be OS signals, these should be saner/safer to use with threads.
If the request only means "don't update the widget", that's easy. Have your threads send the updates to the main program through a Thread::Queue, and then either fetch or discard the log updates from that queue.