I have good flow out of the kitchen faucet, but once I press the button on the spray head, no water is sprayed out from the spray head and very poor flow out of faucet at the same time during the pressing of the sprayer button, then the flow is still poor in the faucet even when the button on the spray head is released. I have to wait for a while to have the normal flow from the faucet, then if you press the sprayer button again, the same thing as before happened. What is going on?

It is probably the diverter valve.
Try Googling for "<the brand of your faucet> diverter valve" and you should find some info that will help.

Here (https://www.askmehelpdesk.com/plumbing/sink-sprayer-cant-find-how-get-diverter-valve-193179.html) is a previous answer of mine with more general info.

Good luck,
WallyH