Hammerschmidt / Lachera - where my Great Grandmothers side of the family originated from?

I am looking for advice please on how and where I would find out where my Great Grandmothers side of my family came from.

The surname is Hammerschmidt. In records they are said to be Prussian Subjects.

My Great Great Great Grandfathers name was Johan Hammerschmidt his wife was Agnieszka Gah possibly Hah, has any one heard of this surname?
My Great Great Grandfather was called Pawel or Paul Hammerschmidt. He married three times and had a total of 13 children.

My Great Grandfathers side of the family we have traced back to then Posen now Poznan. That surname is Lachera.

Kind regards

Karen

Hammerschmidt sounds of course like typical German name, however the Gah name could be Polish if spelled 'Gach' - currently more than 2000 people have this name in Poland. The 'h' letter is pronounced the same like the 'ch' in Polish, and rather rarely the 'h' alone is at the end of a word.

Presently almost 100 persons with Lachera name in Poland:
moikrewni.pl/mapa/kompletny/lachera.html