The CPU cooler should apply reasonably firm pressure on the CPU.
The CPU may wiggle slightly before the retention arm is locked down, but then it shouldn't move at all.
Lastly, do not actively wiggle your CPU, depending on what CPU you have, there are very fine and delicate pins on the CPU (if it's not LGA for example) but there are also equally delicate pins in your CPU socket, any of which get moved or damaged can render the PC unusable or faulty.
You shouldn't have to push hard on the CPU for any reason, just apply the cooler as per the coolers instruction manual. Usually finger tight screwed down is satisfactory.