Yes, it can. Give a call to a local fabricator. He can mix some resin right in your home to match the stone, fill in the missing chip, sand it, buff it, and you're good to go.

If you can't snag a fingernail on it I wouldn't worry about it. If it is irregular in a few areas a good fabricator can patch it so you'll never notice it. what you are seeing isn't common but it does happen with natural stone on occasion. if the crack was proximate to the sink cutout I'd be a little suspicious of the install but given the location it looks like the stone is just being stone. a very thin viscosity adhesive (think crazy glue)can be applied to the crack. It will wick into the stone and prevent the crack from separating. (It shouldn't separate anyway if its properly installed because with proper install there is no mechanical stress on the stone to pull it apart.)

I'm not a Corian specialist, but a clear, 2-part epoxy would be my first choice and you should be able to get it at a hardware store for around $5. Once you are ready to epoxy it, have some tape handy that you can apply to hold the piece in place very firmly while the epoxy sets up. Otherwise, gravity could cause the joint to creep open and you will have a less-than-perfect repair. Even masking tape will work if you apply it well and stretch it a little bit as you bridge the crack. Use multiple layers.

I repaired a cracked toilet tank (crack was on the bottom of the tank) using that liquid rubber roofing and the mesh that is used w/ the roofing material. (The white stuff intended for mobile home roofing repair.) The repair is inside the tank so out of sight. That was probably at least 10 years ago; the repair is still holding. (I have also used the mobile home roof stuff to get more life out of dishwasher racks that a just beginning to rust. In that use, the material lasted close to two years in a dishwasher that was run daily.) So, I would try that approach on the bottom of the sink where it won't be seen. Or open the crack w/ a Dremel on the inside of the bowl and use hydraulic cement. Doing both would be best--hydraulic cement to keep the water out of the crack to keep pressure off the roofing membrane w/ the membrane as a second dam for insurance. But, it's likely that just the cement will do the job.