This is an if/then sentence, which is halting when read. Swap it around and edit a bit:

The concept of cookies is explained in

Needs an ending period:

.Xr nv 9 .

"allows to" is not something generally said.

family of functions obtains the value of the supplied cookie.

These appear to be copies of the dnvlist man page, so I would suggest waiting for that one and using that wording.

s/during/during the/

I replaced this sentence with the one you wrote for dnvlist.

Needs a serial comma after "binaries" to separate it from "arrays":

Returned strings, nvlists, descriptors, binaries, or arrays must not be modified

"in *an* error state."

Typo:
s/rmove/remove/

The sentence ends on the previous line, so this line should be removed.

As above, change from "nvlist_destroy function." to "nvlist_destroy.":

.Fn nvlist_destroy .

and delete the next line,

function.

s/with/with the/

family of functions removes an element of the supplied cookie and frees all resources

(This has been said in this or the other related man page, possibly more clearly. It might be better to copy that one.)

s/If element/If an element/
s/has wrong/has the wrong/

s/will be/is/

"the value associated with the given cookie" like below may be clearer.

It may be clearer to write "must not be modified by the user, since they still belong to the nvlist."

Which nvlist?

consistency: "removes"

the returned memory

What is an "element of the supplied cookie"?

This should be EXAMPLE (capitals).

The one, on which you used functions mentioned above. (next, get_parent, or nvlist_get_pararr) Otherwise you won't get the cookie.

Well, "element connected with the cookie" might be more clear.

Needs an "and"

.Fn nvlist_get_parent ,
and

s/with the/with/ --just "with close(2)."

Please avoid "the following", and just use "this". Also, add "the" to "the cnvlist API":

This example demonstrates how to deal with the cnvlist API.

These should be sorted by section number then name. So:

.Xr close 2 ,
​.Xr free 3 ,
.Xr nv 9

Still needs a new line with and.