If the Court dismisses my civil appeal as of right after a Rule 500.10 jurisdictional inquiry, am I precluded from asking the Court for permission to appeal from the same order or judgment?
No. If a timely civil appeal is dismissed, the appellant may move for permission to appeal. The time for the appellant to make such a motion runs from the service by a party of the original order of dismissal with written notice of its entry (see CPLR 5514[a]); Matter of Park East Corp v Whalen, 38 NY2d 559, 560).