First, make really awfully sure that there is no risk of the run fence itself blowing over ot "leaving", when you make it solid with that plastic. This happens more often than you might think.
If you are absolutely certain the structure can stand the wind load, then yes you can try it the way you propose, but a stronger method would be to use crisscrossed rope or a section of plastic snow fence (= construction fence) instead of the chickenwire. THe drawback of the chickenwire being that it will bag and stretch somewhat, making it a lot harder to get the plastic attached taut (and stay that way) than with the plastic fencing or ropes. (I use the very heavy-gauge baler twine from the 600+-lb roundbales we get for hay, which works very well and is free, but actual storeboughten light rope would work fine too, as long as it is a type that does not stretch much)
Another option would be to go with burlap rather than plastic, for some or possibly even all of the windbreak. I know that the landscape burlap they sell these days seems so flimsy and see-through that it would not do much for wind, but I used it last winter on my turkey run (because of wind-load issues with solid materials) and was astonished at what a good job it did. There was still air movement on the other side of the burlap but not *wind*, it was really quite pleasant back there even on a windy day. They sell it this time of year for people to wrap their evergreen shrubs for winter.
The other other option would be to construct some kind of windbreak *separate from* your run fence, but I would guess you've already discarded this as impractical in your situation. (Indeed, it is best suited to situations where you have access to some cheap or free straw or mulch hay, preferably in big bales and with equipment to use it; or other 'found' materials)
(e.t.a. -- if your plastic is whipping around and flapping itself to death I would not recommend just switching to a tarp. Tarps will do the same thing, albeit not necessarily as soon. Indeed tarps *can* rip apart, or the grommets can rip out, or they can just sort of disintegrate into shreds. I mean, of course you can use a tarp as your plastic but you will still need to attach it more securely across its middle so it doesn't flap, along the lines of what you've suggested or the ideas above.)
Good luck, have fun,
Pat