Is a contractor liable for punitive damages for an injury that was caused bytheir negligence?

It is unlikely that the contractor would be liable for punitive damages because you would have to prove malice and intentional  wrongdoing.  Since this is a negligence case, proving malice and intentional wrongdoing would be unlikely.

As for your negligence claim against the contractor, negligence is based on the exercise of due care (that degree of care that in this case a reasonable contractor would have exercised under the same or similar circumstances to prevent forseeable harm).  For example, a reasonable contractor would have placed warning signs or barricades to prevent injury from the curling twine which would have been difficult to see at night.  Your negligence claim will need to establish breach of the duty of due care, actual cause, proximate cause, and damages.

Breach of the duty of due care occurred when the contractor failed to have warning signs, barricades, etc. to prevent you from tripping on the curling twine at dusk.  Actual cause means but for the failure to erect the warning signs, barricades, etc., would you have tripped and been injured?  If the answer is no, which appears to be the case, you have established actual cause.  Proximate cause means were there any unforeseeable intervening acts which would relieve the contractor of liability?  If the answer is no, you have established proximate cause.

Your medical bills, medical reports, and any wage loss will be used to support your claim for damages (monetary compensation in your lawsuit).  Compensation for the medical bills and wage loss is straight reimbursement.  The medical reports will document the nature and extent of your injuries and will be used to determine compensation for pain and suffering.  Compensation for pain and suffering is an amount in addition to your medical bills.  It is not punitive damages, but compensation for pain and suffering may provide you with substantial additional compensation due to your scar.

In CA, the statute of limitations in a personal injury case is two years.  You will need to file your lawsuit for negligence against the contractor prior to the expiration of the two year statute of limitations or you will lose your rights forever in the matter.

It may not be necessary to file a lawsuit because you may be able to settle your negligence claim with the contractor's insurance carrier.  If that isn't feasible or you are dissatisfied with settlement offers from the insurance carrier, file your lawsuit prior to the expiration of the applicable statute of limitations.

Punitive damages are not available for negligence; they are only available in cases of gross negligence (negligence which shocks the conscience; much more than simple carelessness) or certain willful or intentional bad acts (like assault). Simply stringing twine between stanchions would not rise to those levels, which means you could only collect out of pocket medical costs (and lost wages, if any)--i.e. what's NOT paid by the insurance--and possibly some pain and suffering for the "disfigurment." How much you could collect, and whether you could receive the cost to repair/remove the scar, depends on the facts; you should consult with a personal injury attorney who can evaluate your case in detail. Good luck.