Not for the place as a noun. You would need to call it "windy"

exposed to or swept by the wind:

or "windswept"

open or exposed to the wind:

A place with a lot of wind is called a blowy or tempestuous place.

Blowy: Windy or windswept. [Lexico]

Example: We had a hard time in a blowy place.

Or try blustery.

Blustery: With strong winds. [Cambridge English Dictionary]

A blustery place.

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Dialectically (Dictionary) and rather poetically one might say "wuthering place", if not over-worried about emulating Emily Brontë in her "Wuthering Heights", which has promulgated the term "wuthering" out of dialect towards making it a full English word, although not quite successfully as can be seen from the register "dialectical (British)" that it has retained and the still central association it has kept with the novel.

Three excellent adjectives have already been suggested.

If you need a noun, then you could do worse than the moors or (hat tip to @jtlz2) wold, or talk about the windy [x] where [x] is hills, mountains, peaks or cliffs.

"Wind tunnel" is often uses, somewhat metaphorically, to refer to a windy location.

Of course, technically a "wind tunnel" is a sort of large box through which air is blown in order to provide a controlled testing environment for aircraft parts and the like, but the term might well be used to describe a situation such as one where the wind blows through the space between two buildings. (A little less likely to be used for describing the wind in an open field.)

In nautical terms, there is "the Roaring Forties" - the area between latitudes 40° and 50° south, and known for its strong winds. However, it is chiefly used literally or, rarely, as a metaphor/simile.

The Journal of a Disappointed Man: by W. N. P. Barbellion "January 26. Out of doors to-day it's like the roaring forties! Every tree I pass in the lane was a great wind instrument..."

There is a noun that could be used here, windward. It is usually used as an adjective (windward side, or windward island), but it could also be a noun, meaning "the side receiving the wind's force". With just a smidge of imagination we could extend its meaning to "a windy place".

“We had a hard time on ______(s)”

There is no noun. However, let us suppose you are talking about a journey, e.g.

“We travelled on foot through the mountains. We had a hard time on the windy/windblown/windswept stretches.”

stretch

4a: an extent in length or area

https://www.merriam-webster.com/dictionary/stretch

The high bluffs above the coastline

We had a hard time walking up the bluff above the coastline due to the fierce winds we encountered

Buffeted is another possibility. You might describe "the buffeted hillside", or more specifically "the wind-buffeted hillside". It might not apply as well to an open field. Could work for a coast or a shoreline as well. It can also be used to refer to a place that is buffeted by waves, but in this case you'd usually specify that, e.g. "the shoreline was buffeted by the encroaching sea" or the like. They sometimes go together, e.g. "The rocky shore, buffeted by wind and waves".

"Wind-ridden" is another adjective that can be used to insist on the unpleasantness of the windy conditions in a particular place although this characteristic of the term can be shifted, from applying to a place, to describing a situation associated to a place (example 2 below); as well, the unpleasantness that this word is supposed to communicate (given the meaning of "ridden"), is at times rather elusive (examples 1 and 4 below).

• Love letter to dusty, isolated, wind-ridden … and beautiful Lubbock (example 1)

• Nicole Davis and Hailey Lavarias scored the goals in a wind-ridden affair in Calgary.[…] From kickoff, the wind looked like it was going to be a defining factor for both teams. "The wind made it very difficult for both teams to get the ball moving," (example 2)

• Nor is there a tree in sight, only a great sweep of rolling country, wind-ridden, under a chill sun, an emptiness that leaves no place to hide. (example 3)

• NOW BEFORE HER in the seductive wind-ridden night, the wolf stood in dazzling moonlight. (example 4)

I would suggest "maelstrom". It originally meant a whirlpool but it can be also used figuratively to capture the idea of turbulence, buffeting or chaos.

If the physical location is not important in the context of the sentence, a prepositional phrase such as "in the wind" or "with the wind" may be used as in "We had a hard time in the wind". Using "in the wind" would not necessarily imply that the wind was causing difficulty whereas "with the wind" would.