which sentence is correct?

1) I ate part of the apple pie and chesse cake
2) I ate parts of the apple pie and chesse cake
Both are correct but obviously with different meaning.
I ate part of the apple pie and cheese cake - This is an apple & cheese combined cake (you took only one piece)
I ate parts of the apple pie and cheese cake - There are 2 different cakes, you took one piece each of apple-pie and cheese cake respectively.

Both are correct with different meanings

I ate part of the apple pie and cheese cake means
as a whole, you ate a potion of the pie and cake.

I ate parts of the apple pie and cheese cake means
the pie and cake were cut or broken in pieces before and you ate the bits and pieces from the crumbs

Hi,

I ate (a) part of the apple pie and CHEESE cake.

Note:
1) You have misspelled the word " cheese".
2) " a part" = a little bit of something.( a small bite or something)
3) part = a bigger piece of something. ( a bigger part . It is bigger than a small bite.)

Hope it helps you.

Ciao

I ate part of the apple pie and cheese cake.

No need to add 's' after 'part'

I ate part of the apple pie and chesse cake

I ate part of the apple pie and cheese cake


cheese not chesse ok

I ate part of the apple pie and chesse cake