You CAN make it work with just the chicken wire...especially on a budget. I would go hardwire cloth if you could afford it....BUT....depending on the predators in your area, you might not need it. In my situation, they will be locked up tight at night, and there will be no access for predators (door has deep lip and can't be lifted, and 2 windows have multiple locks..i.e. gate latch and barrel bolts). During the day they will be in the run, and I will go with chicken wire. 1. because it is less expensive, and 2. The only true predator during the day would be hawks in my area (and that is quite sparse)...and I have a dog that would keep them at bay anyway. Coons and coyotes would have to scale my 6 foot fence, and in the daytime, you aren't likely to see coons running around, and I am in a neighborhood, and I haven't seen a coyote during the day. Another reason I won't use the hardwire in the coop windows. They won't utilize the coop during the day except for egg laying, and we don't have predators that are going to get into the coop during the day with our dog providing security...and at night they just won't be able to get in..so I don't want to waste the money(hoping I don't learn a hard lesson though...lol)
If you can overlap the wire to close down on the hole size a bit it will help. It is all about how much risk you are willing to take, and your comfort level.