Sealing a sprayer faucet leak?
It also looks like there is a small stream going down the spout towards the handle. My suspicion is that there is enough internal corrosion that you will likely, even if you seal all the current leaks, develop more in short order. If that was my home I would replace it. If cost of replacement is an issue, I would guess that some epoxy (J B Weld or similar) would be an ugly but functional temporary fix, until more corrosion eats through a different area.