Ne vs neyi?
When do I use ne and when do I use neyi

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I'm learning and battling with this as well. Is not ''neyi'' the accusative form (direct object form) of ''ne''?

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Neyi is the accusative form of the word Ne

some examples;
- Neyi beğenmedin? What haven't you liked?
 - Bu arabayı beğenmedim. I did not like this car.

***There are some verbs that requires to add accusative suffix ı/i to the objects in the sentence. And beğenmek is one of them. And there are some verbs that is okay to use with or without this suffix- when you use it works like "the" in English, when you don't add, it means the object is not certain.

Çocuğu gördüm. I saw the kid.
(Bir) Çocuk gördüm. I saw a kid. ---> either you say bir or not the sentence means the same.

http://blog.learnturkisheasy.com/2012/10/neden-why-neneyi-what.html

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NE:

She picked flower for her mum. ( O, annesi için çiçek topladı.) Ne topladı? Çiçek.(Flower's mission in Turkish grammar no symptom object)

NEYİ:

She gave flowers to her mum. ( O, çiçekleri annesine verdi.) Neyi verdi? Çiçekleri.
(Flowers's mission in Turkish grammar symptom object)

Answer

actually same things :) 
but neyi is more specific ....