For cases in NYC, if the defect that caused your accident is on the sidewalk and if it is in front of a single or two family home, the City of NY is liable for your injuries, but only if there is “prior written notice”. Prior written notice means that the City of NY received from someone at some time, more than 15 days prior to your accident, written notice about the defect and the City failed to repair it.
The same applies to cross-walk cases. If the defect is in front of anything other than a single or two family home (apartment building, commercial space, etc.), then the private land owner is liable for your injuries, provided that we can prove that they were negligent.