I have a peerless 2 handle kitchen faucet with sprayer- no water from anywhere. I took it apart knowing I have hard water, cleaned the aerator, cleaned the two part plastic valves in each handle. I used vinegar to run through the line from side to side. Putting it back together I checked that the water was coming out of the handle part before the handle valves were put back on and I did get water. Still no water. I guess the problem is in the middle where the spout attaches, inside it looks like a round metal piece in the center with a plastic washer of some kind with the middle cut out (hex-shaped where the metal round stem is). How do I even remove this and check or replace? Does it come out with a large hex tool? Thank you for helping.

I checked that the water was coming out of the handle part before the handle valves were put back on and I did get water Are you saying that you took the stems out and turned the water on to flush out the supplies. Did you get pressure out of the supplies? Back to you, Tom