My kitchen sink has a faucet with a single handle. I also have a pull-out hand sprayer next to the faucet, which is connected to the faucet by a hose underneath the sink. Typically, after I use the hand sprayer, the faucet does not work. I can move the faucet handle up and down and sometimes that will work and get the water flowing. Now the faucet won't turn back on at all. It's a Chicago Faucet.

Wow, very interesting. Sounds to me like something wrong in the faucet valve itself. Is it an older faucet? It would be a fairly inexpensive fix to replace it. Or if you're a curious type then you could always take apart the entire faucet assembly and inspect it for worn out bushings or ball ball valves.

If you can disassemble the faucet you should find a SPRAY DIVERTER (located at back side behind spout) that will need to be cleaned or replaced. You will literally need to lift the spout of the faucet off the body of the faucet to see it. I have attached some pictures of a similar spray diverter for reference.

Once you get the diverter out try cleaning it and then reinstall it. If that doesn't fix the issue the bring the diverter over to a local plumbing supply store and they'll order a replacement part for you.

Questions? Let me know, OK?

Mark