Consolidated Edison says the hardest-hit areas are Yonkers and Mount Vernon. Service may not be fully restored in some areas until Thursday.
Bronxville, New Rochelle and Eastchester also were affected.
A customer can be anything from a single-family house to an apartment building, or a business.
Long Island had about 5,320 outages at one point, mostly in Nassau County.
The Long Island Power Authority said the hardest hit areas were along the north shore of Nassau County, and parts of central western Suffolk County.
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