The ship Viva was attacked by a drone, the Ukrainian Navy writes on social media, adding that none of the eleven crew members were injured.
A video from the Navy shows the damaged ship with water on the deck.
The ship, which was loaded with sunflower oil and could continue its journey towards Egypt, was in Ukraine's economic zone, in the so-called grain corridor that is supposed to be safe for maritime traffic.
The attack is the second against a Turkish ship in the Black Sea in as many days. On Friday, a fire broke out on a ship in connection with a Russian airstrike on the Ukrainian port city of Odessa.
On the same day, Turkish President Recep Tayyip Erdogan called on Russian leader Vladimir Putin in a meeting to agree to a ceasefire regarding energy facilities and ports.
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