Texas:
You can try your hand at making real plaster (which is a mixture of hydrated lime, Plaster of Paris and sand) but the only time you fix walls with real plaster nowadays is if you're repairing a historic building where it's important to be as true to the original building as possible. In that case, you'd use the same materials and methods that were used by the original builders. I once tried making and repairing walls with real plaster, and it worked, but I found it very hard to work with.
Years ago, people would make plaster by mixing Calcium Oxide (CaO, which is also called "Quicklime") with water. That is a highly exothermic reaction so be extremely careful if you ever try making real plaster from Quicklime. When that slurry cooled it was called "lime putty" and you can buy the same stuff nowadays in powder form as "Hydrated Lime". You mix hydrated lime with Plaster of Paris (but I forget the mixing ratio) and this will give you the "gauging coat" which is the white surface plaster you see immediately under the paint. To make the base coat plaster, which is the stuff that goes on first, you simply mix fillers into gauging plaster. Typically, they'd add up to 10 parts sand per part gauging plaster, but in much older buildings they'd even use horse hair as a filler material for the base coat because it helped hold the base coat together.
Real base coat plaster isn't very sticky and if it has lots of sand in it, it can be very heavy, and so it took some experience to mix real plaster that would stick well to lath without falling off.
Nowadays, the base coat plasters that are available have powdered glue added to them so they stick much better to everything, AND they use perlite as the filler material, which is very light. (I've even seen shreaded newsprint in base coat plaster.) So, applying base coat plaster nowdays is a no brainer. Of the two modern base coat plasters I've used and am familiar with, which are Perlite Admix Hardwall from Domtar and Structo-lite from CGC (a subsidiary of USG (the US Gypsum Company)), I much prefer the former; the Perlite Admix Hardwall. It comes in a big bag that costs about $15, and has shreaded newsprint in it.
Nowadays, no one uses guaging coat plaster anymore either. They use drywall joint compound over the base coat plaster. There are premixed and powder forms of drywall joint compound, and the little experience I have with the premixed stuff is that it's awfully thick and needs to be thinned out with water to make it spreadable. I think that's because neither the manufacturer nor the customer wants to pay extra for shipping water. The manufacturer knows the customer can thin it as much as he likes at the job site.
Both premixed and powdered drywall joint compounds come in three different kinds:
1. "Regular" or "Taping" joint compound which has the most glue in it so that it sticks the best, but dries the hardest and is therefore the most difficult to sand smooth.
2. "Finish" or "Topping" joint compound which has the least glue in it so that it dries the softest and is the easiest to sand smooth, and
3. "All Purpose" which has some glue in it and is a compromise between adhesion and hardness so the contractor doesn't have to carry two different bags or boxes around everywhere in his truck.
There are different ways to repair plaster walls, but the simplest would be to simply mix base coat plaster with water and moosh (pronounced "press") it onto the expanded metal (wire mesh) with a putty knife or plastering trowel, trying to get it reasonably flat and smooth. When that's dry, then you cover it with a drywall joint compound, and I'd use an All Purpose joint compound.
A better way in my view would be to cut the damaged area of the plaster wall out and fasten drywall to the wall studs. Paint that drywall with white wood glue diluted with enough water to make it into a paintable consistancy, and let that dry. Then, apply your base coat plaster to the drywall to build up the wall thickness to within 1/8 inch of the surrounding wall. The moisture from the wet base coat plaster will re-activate the dry wood glue, effectively gluing the base coat plaster to the drywall. Allow the base coat plaster to dry. Then, apply drywall joint compound over the dry base coat plaster to serve as the gauging plaster. If it's got plaster walls it's an older building and you may have a thick accumulation of layers of paint on the wall. If possible, remove some of that paint around the repair and put fiberglass mesh drywall joint tape between the old wall and the repair. That will prevent cracks from forming around the perimeter of the repair later on.
You should be aware that expanded metal (which you're calling wire mesh) will only be used in certain areas within plaster walls. Typically the inside and outside corner beads in plaster walls will have expanded metal flanges on each side, so that might be what you're seeing. Also, window "returns" on plaster walls will typically be done with expanded metal. In a house, the exterior walls are typically only a bit thicker than the studs used to frame those exterior walls. However, in larger buildings like apartment blocks, the exterior walls will often be made of concrete blocks with a brick veneer on the outside and 2X2 wood strapping and 3/4 inch thick plaster on the inside. That assembly can be 14 inches thick, and no window frame is gonna be that wide. So, the plaster or drywalled area between the corner bead surrounding the window and the wood window frame itself is called the "return". On older buildings with plaster walls, they'd typically just staple or nail expanded metal over the returns and apply the base coat plaster to that expanded metal.
So, don't expect to find expanded metal everywhere in your walls. The walls will normally have base coat and guaging coat plaster applied to wood or gypsum board lath. Expanded metal will only be found near corners (as part of the corner bead) or around windows in walls that are thicker than the window frame and that therefore have "returns" to accomodate that difference in thickness, and in any area where it would have been difficult to apply wood or gypsum board lath.
Building up the base coat plaster almost flush with the surrounding wall is the hardest part, and if you look at old plaster walls from the back, you'll often see short pieces of wire. They used to use short pieces of wire to hold wooden or metal "screeds" to the wood or gyproc lath to get their base coat plaster on in a uniform thickness. You can do the same by using double sided tape to stick short pieces of wood to the drywall to be used as screeds to achieve a uniform thickness of base coat plaster. Once that's dry, you can remove the wood screeds and fill in the resulting slots with more base coat plaster.
Another way to get the base coat plaster on would be to use your trowel to get it flush with the surrounding wall, allow it to dry for a couple of hours until it's stiff, but not dry. Then wipe it down with a damp sponge to remove the surface of the base coat plaster until you remove enough to provide space for the top coat of joint compound. I've repaired plaster both ways, and I find the wiping down (with the sponge) method is faster and generally easier to do, and provides just as good results.
And, as with all wall repairs, work with a bright light shining on the work area at a sharp angle. That will exagerate the roughness of the surface, but will also allow you to judge much better where to add material and where to remove it to get the wall smoother and flatter. When it looks OK under critical lighting, it'll look perfect under normal lighting.