In the Appellate Division, First and Second Departments, the short answer is no. A Record on Appeal comprises the documents that were before the Court of original instance when an Order being appealed was decided. Thus, if the documents that were determinative of that Order did not include the new evidence the Appellant now seeks to include in the Record, they are not properly included in the appellate record.
What the attorney should have done, if time permitted when the new evidence was discovered, was to file a Motion to Renew, returnable in the lower Court. If that motion had been granted, the problem would have been solved, and the appeal would have become moot. If the Motion had been denied, the attorney would have been able to include the motion papers from the renewal motion in the Record on Appeal, thus accomplishing the initial goal.
Having failed to make a Motion to Renew in the lower Court, the attorney has only two options. If the other party is willing to stipulate to the inclusion of the new evidence, it is unlikely that the Court would object to its inclusion in the Record on Appeal. However, this is no guaranty. The only definitive way to get the new evidence into the Record is to make a Motion returnable in the Appellate Division to expand the scope of, or enlarge, the Record on Appeal. These Motions are seldom granted, but under certain circumstances, the Court may indeed grant leave to include the new evidence in question. If the Motion is granted, simply include the new evidence in your Record. If not, the inclusion of said documents is not permitted.