Any part of the circuit that stores a bit of memory needs a clock to tell it when to store the input value. Both the program control unit and the register use a flip-flop, so they need the signal from the clock.
That signal goes 0, 1, 0, 1, 0, 1, and on, and on. It does this at a steady rate, called the clock rate. The clock rate tells you how often per second the clock shows a 0 and a 1. If the clock rate spends half a second being 0 and then flips to 1 for half a second, the clock rate is 1 Hz, because the clock will show each value once per second. If the clock rate spends a quarter being 0, then flips to 1 for a quarter second, then back to 0, and back to 1, the clock rate is 2 Hz, because the clock shows each value twice per second.
The 1-bit processor runs at 0.5 Hz, so it spends one second being 0 and one second being 1. Now, don’t you accuse it of being slow! It could do 10 Hz, or 1000 Hz, or even many MHz, but you wouldn’t be able to see what’s going on. It’s just trying to be helpful.