Whose truck it was, and why it is often not your contractor's
This is the part that surprises people. On a concrete job the heavy vehicle that broke the curb is almost always a ready mix truck, and a ready mix truck belongs to the batch plant. Your contractor ordered the load and was standing there when it happened, but the truck, the driver and the company behind both are the supplier's.
That does not make your contractor a bystander. They scheduled the delivery, they know which truck it was and what time it came, and they are the fastest route to the plant. It does mean that when somebody asks who fixes it, the answer runs to the company whose truck it was rather than to whoever happens to be holding a trowel.
The other version is simpler. If it was the crew's own pickup, dump trailer or equipment trailer that clipped it, then it is the crew's, and the conversation is short. Either way the useful move is to sort out which of those two you are looking at while everybody is still on the property and can see the same thing you are seeing.
Whose curb it is decides what an acceptable repair even looks like
A curb at the edge of a public street is usually not your concrete, even though it runs across the front of your lot and you are the one looking at it. It generally sits in the right of way, which means the locality has an interest in how it gets put back and what it gets put back with.
That matters more than people expect. When a curb belongs to a town or a county, a repair is not finished when it looks acceptable to you. It has to match what they will accept, and some localities want that work done by someone already approved to work in the right of way. Patching it yourself over a weekend can turn one problem into two.
Whether your county or town wants a permit before anybody works in the right of way is a five minute phone call, and our Virginia permit guide is the place to start on who to ask. Make that call before the repair, not after it.
Keeping the trucks off it costs far less than putting it back
Here is the fork that is actually in your hands. Before a pour, you decide where the loaded trucks go, and a curb gets broken when a mixer crosses it at an angle. The wheel loads land on an unsupported edge and the corner shears off, or the whole face cracks back and the crack shows up later.
One option is to let the truck cross, and to do it as well as it can be done. That means crossing square rather than cutting the corner, crossing at the driveway apron where there is a proper transition, going straight in and straight back out with no turning under load, and one truck at a time.
The other option is to leave the truck on the pavement entirely and move the concrete instead. That is our answer on most jobs where the entrance is tight or the curb is worth protecting. We run a trailer mounted line pump on 2.5 inch and 3 inch line, so the mixer backs to the trailer pump's hopper at the street and the concrete travels the rest of the way through hose. Hose weighs what two people can carry and it goes over a curb without noticing it is there.
Neither option is free. The first spends a little risk on the curb. The second spends a little planning, because the trailer needs solid ground and the hose route has to be walked ahead of time. What tips it is usually how much the curb would cost to put back properly, which on a public curb is more than most people guess.
Patching the face against cutting the section out
When it is already broken there are two real repairs and they are not equivalent. A patch bonds new material onto the damaged face. A replacement cuts the broken part out and pours a new piece of curb in its place.
A patch is quick and it is reasonable for a small chip out of the face where the top edge is still intact. What works against it is where the damage sits. A curb edge takes wheel hits, road salt, a plow blade in winter and freeze thaw at this elevation, all of it working on the one line where old concrete meets new. Patches on a curb nose tend to come loose eventually rather than fail all at once.
The replacement is the version that lasts. The section gets saw cut back on both sides to a clean straight face, the broken concrete comes out, the base under it gets checked and put right, forms go back and a new piece gets poured, doweled into the sections either side so the panels move together instead of independently. It costs more and it takes a couple of visits with the cure in between.
Be ready for the colour. A new curb section against weathered concrete reads as a different piece, and it will not close that gap with age. Cutting to a clean joint at least makes it look like a panel that was replaced on purpose rather than a scar somebody smeared over.
What to do the day it happens
Say something before the trucks leave. That is the whole trick and almost everyone misses it, because on pour day there is concrete going down and a broken curb feels like tomorrow's problem.
Take photographs while the truck is still there, with something in frame for scale, and note the time and which truck it was. Tell your contractor and ask them to raise it with the plant that day. If the curb is in the right of way, report it yourself rather than letting the locality find it themselves later.
Do not have anybody smooth a patch over it that afternoon to make it look better. Once the damage is covered up, the size of it is a matter of opinion, and you have narrowed what can be done about it. Leave it as it is, get the pictures, and make the calls.