IS the kitchen faucet the kind where you grab the arm and pull it out and use it as a sprayer. Or the kind with a sprayer off to the side.
In that order
At teh "Base" of the pull out part (Where the hose connects) is some kind of backflow preventer. THis may have "Gunked up" and needs cleaning.. A vinegar boil may do it. I don't really like the chemical cleaners. or get a new back flow preventer.
UNSCREW teh hose and it should be right there.. You may be able to tap-tap-tap it out.
On the 2nd
IN the faucet base (You will need to remove the faucet arm is a "DIVERTER" valve. Same thing.