He contempts the work.

Contempt is a noun. We can say {show a contempt for sth.}

He shows contempt for the work.

I jump on the table.

Good!

They despise to eat hamburger.

despise 套现/很不喜欢 (动词）

They despise hamburgers.

I run around in my house.

Good!

A dog barks at me.

Good!

We cook some cookies.

We always say {make some cookies}.

We make some cookies.

A chicken runs to farmer.
{Farmer} is a countable noun. You should say {a/the farmer}.

A chicken runs to a farmer.

We admire cars.

She cherish her cat.

{She} is the third person singular, so please add {-es}.
第三人称单数作主语,其后动词以-s/-x/-sh/-ch结尾,要加es.

She cherishes her cat.

We jog to our home.

Good!