How to backflush kitchen sink with weird check valve (crude diagram included)?

[eyc]

Recently had a hot water heater replaced, which exacerbated a problem I've always had with my hot water lines: calcium/lime deposits. Thin egg-shell-like, pastel-green colored, flakes in the pipes.

Usually not a problem. I clean out the lines at the clogged faucet and it's good for several months. But, now I'm having trouble at the kitchen sink and I'm hoping I could get some help. (I'm a complete amateur when it comes to plumbing, so please forgive the lack of proper terminology.)

Relevant facts:

1. Kitchen faucet is a basic Moen single-handle pull-down faucet from Home Depot (Link to HD here: Moen "Indi" Faucet).
2. Hot water pressure at the faucet was low, so I did my normal routine.
   1. Turn off hot water valve, disconnect at Point A (see diagram), and pick out flakes from the spigot and tube.
   2. Put faucet handle in neutral and turn on, which back-flushes cold water out of the hot water tubing.
   3. Reconnect, and do the same thing on the cold water side for good measure.
3. This resolved the hot water pressure, but then I noticed that both lines had lower than normal pressure coming out of the sprayer.
4. So I cleaned out the sprayer at Point B and I disassembled the ball-valve at Point C and flushed out all pathways.
5. But, the pressure was still low on both lines, so, I decided to check the main output line that has the black weight attached to it.
6. What I realized is that the line (with the black weight) is connected to the rigid tube coming out of the ball-valve with some kind of quick connect junction (Point D).
   1. When I disconnected that quick-connect and looked up into the quick-connect junction, I saw that it had some kind of check-valve inside it. A white, plastic, mechanical structure that completely blocks the view up the tube. Upon closer inspection, I could see tiny fragments of the calcium deposit (like sand) trapped around the inner perimeter of the valve.
   2. I have a suspicion this is where my blockage is.

Why is there a check-valve in this quick-connect thing? What purpose does it serve? More importantly, is there some way to back-flush that valve? I tried to connect a cheap plastic tube directly from Point A to Point D and turn on the hot water, and it was a disaster with water spraying everywhere. Now that I drew up the diagram, I see the error in my ways -- there would be nowhere for the hot water to go unless I turn off the cold water and then disassemble the ball-valve at Point C (which I didn't do). Even then, if this actually is a check valve, would water even travel backwards through that valve?

Unfortunately, I cannot get rid of the quick-connect thing at Point D. It looks like it is integrated into the rigid tube that comes out of the ball-valve.

Any help would be greatly appreciated.

Edit: I found a picture of the quick-connect thing, directly from a Moen schematic:

 

[Jeff Handy]

Moen will send you a new one.

And they may have several choices of flow rate.
Ask for the highest gpm flow rate.

 

[Jeff Handy]

The check valve is there so water from the faucet head can’t get sucked back into the house plumbing.
If the spray head was laying in dirty water and house water pressure dropped, causing a vacuum.

 

[eyc]

SOLVED: I pulled on the check-valve inside the quick connect fitting. To my surprise, it backed right out of the fitting, allowing me to clean it. (Pictures below.) As you might imagine, it was completely packed with deposits. A rinse of that valve and the water pressure feels like it doubled.

 

[Jeff Handy]

You also should flush your supply hoses and shutoff valves once in a while.
Those flakes will build up there, and will enter the faucet when you are operating the valves.