Yes you can fix this, and since we are all stuck at home, why not give it a shot. Is that leak coming from the drain? I'm not sure what a popping is, it looks like the drain assembly is leaking, but if the water is coming from up at the faucet where your camera goes at the end then you can leave the drain alone. First thing is buy a new faucet. It's just not worth messing around with that faucet since it doesn't look expensive. Also replace the supply lines. You can bring them to a hardware store or plumbing supply and they'll give you new ones. If not you can order them online, you'll have to check your supply size, but it's very likely 3/8ths. These should be changed every 5 years or so anyway. They could last 5 years, they could last 30 years, but if they burst you could be in for 10s of thousands in water damage. Take this moment to think about what other supply lines in your house you might want to change out. stainless braided are usually fine, it's not worth cheaping out here. Washing machine hoses receive a lot of stress so consider changing those, and maybe adding hammer arrestors if you don't have them.
Install the new faucet per the manufacturers instructions. Now if the drain is still leaking, I would take it apart and rebuild it where it is leaking. You could buy some new washers for the slip joints if that is where the leak is happening, but for a drain like this tightening it up may work.
Read the manual for the faucet you buy so you know what materials and tools you need before you start, and watch a youtube video on drain assembly. It sounds scary, but honestly none of this is very hard. If you can assemble Ikea furniture, in my opinion you can do this. Good luck.
DIYorBust