Jen, first of all, your ancestor did not come from Hanover, Prussia, because the Kingdom of Prussia (in German: Preussen; capital: Berlin) did not annex the Kingdom of Hanover (in German: Hannover; capital: city of Hanover) until 1866, at which time Hanover became what was to then remain until 1945 the Prussian province of Hanover.But your ancestor was born and left before Hanover became part of Prussia.
Vital records are not and never have been kept at either the national, state, or provincial level in Germany.In Germany, vital records are kept strictly at the local level.So you will not be able to proceed until you do additional research in the U.S. and find out just which city, town, or village in Hanover your ancestor was from.
German churches in the United States tended to keep rather detailed records.It is thus quite possible that August Philipp Winter and Friederike Kessler's CHURCH (I emphasize church) marriage record would name the town in Hanover where August was born.So your next step should be to obtain a copy of that church marriage record.
Here is the link to a map of pre-1945 Hanover, so that you know what it looked like.(Hanover is in the purplish color.)Hanvoer covered an area roughly equivalent to that covered by the U.S. states of Maryland and Delaware combined.So like I said, it's impossible for you to proceed with Hanover research without knowing which city, town, or village in Hanover August was from.
Following World War II and the break-up of the huge state of Prussia, the Prussian province of Hanover, the state of Oldenburg, and the two very small states of Brunswick (in German: Braunschweig) and Schaumburg-Lippe combined to form today's German state of Lower Saxony (in German: Niedersachsen), with the city of Hanover (in German: Hannover) as its capital.