Thanks for the tip, John.
I shut off the water then removed the cap, but I didn't know what to remove next. I didn't see any flat edges (which would indicate where to grip with a wrench) nor any other obviously movable parts, and I didn't want to force anything and cause more damage. I don't see a lock nut anywhere.
In this photo, I think the white piece is what you recommended to replace. How should I proceed from here? There are two notches in the top of the white piece, but I couldn't rotate it with hand pressure.
I cleaned some sediment that had accumulated on the underside of the cap. After I reassembled everything, the water flow decreased significantly - it reduced to about 25% of what it was before this procedure - so I think jostling the white piece had some effect. The sprayer's flow was not affected.
I included a shot of the underside, just in case that might be useful.