My bike has front and rear mechanical disc brakes, and the front wheel is QR while the rear is fixed. My rear brake is doing fine and working fairly well. However my front brake has been behaving strangely recently. Whenever I pull the lever quickly, the brake will "grab" or "clip" the disc later (the distance between the lever and the handlebar is less than when I pull the lever slowly) and the brake sqeals really bad. If I pump the lever a few times before fully engaging or if I pull the lever slowly and gradually, the brake works perfectly without any squeals. Can this be the cable slack, with that being said a cable replacement would improve the situation? Any help is appreciated, thanks!

cannot see what you have as an issue. (or visualize it from your description)
But, as it is , you have to post a picture somewhere else, then post that address ..

since posting pictures, minimum , Here,
takes 10 times that you have posted/ replied here first, not just one.


FWIW, I recently fitted a road disc caliper*,
on a bike that had a mechanical disc MTB caliper,
left the Long (adjustable, Avid Speed Dial ) pull levers on
and it's been fine , slack taken up by the time the pads hit the disc..

* TRP Hy Rd, Which only comes as short cable pull..





...

I don't understand the idea of pumping mechanical disc brakes or how the speed of pulling the lever could have an effect.

To me, unless something is damaged, this sounds like the pads, and maybe the caliper, need to be adjusted to compensate for pad wear.

There is certainly a youtube video that will cover this.

What disk brakes? Do both pads move or just one?

Sounds like something is loose somewhere.

check the cable tension, the space between the disk and pads, the screws holding the brakes and check to see if anything is stopping the lever or calipers.