Will homeowners policy cover replacement of driveway culvert?

It all depends on whether or no the driveway culvert is covered by a covered peril listed in your policy and that there is not an exclusion which would then exclude coverage. For example if your policy has the peril of "vehicle collision" in it, and a vehicle collided with the driveway culvert and caused the damage, it would be covered as long as there is not an exclusion saying it would not be covered for some reason. Also, you would have to own the driveway culvert and not the city, as your policy only property you own.