Alan,

I searched
https://genealogy.cjh.org/family-collections
for their NYC incorporation papers using search term Hazedek.  The incorporation papers should provide you with lots more information.
You can see all the search results here:
https://genealogy.cjh.org/fcresults.php?keyfilter=any&inputtext=Hazedek&inputtext2=&searchtype=words&dbselect=all&Search=Search

Name: Congregation Linas Hazedek B'nai Menasche
Year of Incorporation: 1911
Document Number: 4073 Microfilm Reel: XXXIII
Collection: I-154: Incorporation Papers (1848-1920)

Name: Congregation B'nai Menasseh (hungarian) Linas Hazedek
Year of Incorporation: 1889
Document Number: 110 Microfilm Reel: XII
Collection: I-154: Incorporation Papers (1848-1920)

Name: Congregation Nachlas Zvei Linas Hazedek Bnai Menasche Year of Incorporation: 1893 Document Number: 253 Microfilm Reel: XV Collection: I-154: Incorporation Papers (1848-1920)

Name: Congregation Nachlas Zwie Linas Hazedek B'nai Menasche Year of Incorporation: 1911 Document Number: 4073 Microfilm Reel: XXXIII Collection: I-154: Incorporation Papers (1848-1920)

Name: Congregation Nachlas Zwie B'nai Israel Linas Hazedek B'nai Menasche Year of Incorporation: 1911 Document Number: 4073 Microfilm Reel: XXXIII Collection: I-154: Incorporation Papers (1848-1920)

Regards,

Sherri Bobish