Over time, sink sprayers often break or become clogged with mineral deposits. Or the sprayer hose can harden and crack or wear through from rubbing against something under the sink. The best solution in these cases is replacement. You can pick up just the sprayer head ($5) or a head and hose kit ($10) at a home center or hardware store.
Photo 1 shows how to remove the entire sprayer head and hose assembly. You may be able to get a small open-end wrench up to the sprayer hose nipple, but space is very tight. If there isn't enough room to turn the wrench, you'll have to purchase a basin wrench ($15 to $25 at home centers and hardware stores). If your sprayer hose is in good condition, simply unscrew the head and replace it (Photo 2).