If my son received a deep puncture wound under his chin on an exposed bolt at a city park. is there anything I can do to hold the city responsible?

You should contact the city government and find out the procedure for filing a personal injury claim against the city.  The city may require you to file an administrative claim which will probably be denied before filing a lawsuit against the city for negligence.  If you don't timely file the administrative claim and miss the filing deadline, your subsequent lawsuit against the city for negligence may be barred.

When you contact the city government, in addition to their procedure for filing a personal injury claim, you also want to find out the name, address and telephone number of the city's insurance carrier. 

When your son completes his medical treatment and is released by the doctor, obtain his medical bills and medical reports.  (I assume that your son is quite young and there isn't any wage loss claim; otherwise documentation of wage loss would also be included).  Your son's personal injury claim filed with the city's insurance carrier will include these items (medical bills, medical reports and if applicable documentation of wage loss).  Compensation for the medical bills is straight reimbursement.  Compensation for wage loss is straight reimbursement.  The medical reports will document the nature and extent of your son's injury and will be used to determine compensation for pain and suffering which is an amount in addition to the medical bills.  If the case is settled with the city's insurance carrier, NO lawsuit is filed.  If you are dissatisfied with settlement offers from the city's insurance carrier, reject the settlement offers and file your lawsuit for negligence against the city (after having complied with any applicable administrative claims procedure).  You will need to be appointed guardian ad litem to file a lawsuit on behalf of your son if he is a minor because a minor cannot file a lawsuit himself.  If the case is NOT settled with the city's insurance carrier, you will need to file your son's lawsuit for negligence prior to the expiration of the applicable statute of limitations or your son will lose his rights forever in the matter. 

You could try to sue the city, but bear in mind:

1) The city would have to be at fault in some way--that is, it's not enough that the bolt was exposed, but it would have to be considered negligent, or unreasonably careless, to have that bolt exposed and the city would have had to have had some warning or notice of the issue.

2) Suing a municipality is always more expensive and difficult than suing a private party--there are special procedural rules which have to be complied with.

3) If  you son has (hopefully) not suffered some long-lasting impairment, disability, or disfigurement, all you could likely recover are your out-of-pocket (i.e. not covered by health insurance) medical costs.

It is often not worthwhile suing a municipality, unless there are significant costs or losses involved.