He runs to the school.

Good!

We squeezed the lemon.

Good!

She cuts the pizza.

Good!

I despise vegtable.

Watch your spelling! Vegetable! Also use plural form please.

I despise vegetables.

They contempt those work.

{Contempt} is a noun. We always say {show contempt for something}.

They show contempt for those work.

We hate to do the work.

Good!

I adore hamberger.

Watch your spelling! Hamburger! Also, {hamburger} is a countable noun.

I adore hamburgers.

A cat hides its things.

Good!

We slide down the Park Slide.

Good!

He cherishs the food he had.

The third person singular form of {cherish} is to add -es because it is ended with -sh.
Keep the same tense here.

He cherishes the food he has.
He cherished the food he had.