Usually the diverter valve is a small piece unside the faucet assembly.
We had a similar problem with our kitchen sink spray nozzle a couple of years ago.
I replaced the diverter valve with no improvement.
Turned out to be a simple problem. The rubber sealing washer between the end of the hose and the spray nozzle had swollen so much that it was strangling the flow. All I had to do is ream the center hole in that washer larger and "Bob's your uncle". (Too bad I can't fix my 69 year old prostate gland as easily myself.)
Try looking at the spray head first, It's usually removeable ftom the hose by unscrewing a knurled fitting by hand. If you can get a good flow from the hose when you open a faucet, the problems in likely in the spray head otself, maybe something as simple as that swollen washer in ours.
HTH,
Jeff