It is a Royalex canoe, correct? As far as I am aware, the Tripper was only made in Royalex. It looks like an old Kevlar felt skid plate. This is exactly what happens with Kevlar felt plates, hard impacts cause sizable pieces to break off in irregular chunks. With Kevlar felt the individual fibers are strong, but very short, so they provide minimal overall structural integrity.
I am assuming that you do not want to try to remove the entire existing plate and make a better one, which is what I would probably do. If so, your best bet is to get some West Systems G Flex epoxy which is formulated to provide better adhesion to plastics such as the ABS of Royalex. Look around for the West Systems 650-8 package, which has 4 ounces of unthickened resin and 4 ounces of hardener, which should be enough to do the repair. This can usually be found from multiple vendors for $20-25. You will also need some plain weave fiberglass cloth in 6 ounce/square yard weight.
Start by going around the defect in the plate and beveling the edges of the defect to a smooth bevel up onto the intact plate all the way around. Make the angle of this bevel around 30 degrees or so to the surface of the plate surface so as to provide a nice, wide bonding area for your repair. As you do this if you find any portions of the plate that are not well bonded to the hull you might as well remove them as they will likely come off later, along with a portion of your repair.
Your bond to the hull will theoretically be better if you can remove the vinyl color layer of the Royalex of the hull exposing the underlying ABS which will be a different color. You can do this with coarse sandpaper but it takes some time. The bond strength is also enhanced somewhat if you flame oxidize the surface of the ABS before you apply the first layer of fiberglass cloth. This is done with an inexpensive, hand-held propane torch and there are detailed instructions on exactly how to do this that comes with the G Flex epoxy. Flame oxidation is done right before you apply the first layer of cloth. When you start your repair, it is best to mask around the entire area of repair with masking tape.
Once you have the surface prepared, make a template of the defect extending all the way out to the limits of the bevel you created on the Kevlar plate out of stiff packing paper. You will use this to mark your cloth. Lay out the fiberglass cloth on a flat, clean surface making sure you keep the lines of the warp and weft of the cloth aligned at right angles. Put your paper template on the cloth and mark it with a sharpie a little past the edge of the paper. Carefully cut the cloth out with a scissors handling it gently to minimize fraying. Mix up a fairly small batch of epoxy which can be done by eye using equal volumes of hardener and resin. If you need more epoxy during any one step you can quickly mix up another small batch. Apply a thin layer of mixed epoxy over the entire area of repair, then carefully lay your first layer of cloth on and fully wet it out. You can apply additional layers of cloth over the first as soon as the epoxy on the first has become tacky enough that you won’t disturb the underlying area.
The idea is now to apply more layers of fiberglass cloth the same shape in slightly smaller concentric sizes which will build up and replace the Kevlar material that you removed when you beveled the plate. You can use the same paper template and trim off just a couple of millimeters all the way around for each new patch. You will wind up with a defect in the center of uniform depth that you can then fill in with patches of nearly the same size and shape. The idea is to fill in the hole with as many layers of cloth as it takes to build it up to or just above the level of the surviving Kevlar plate. Once this is accomplished and the epoxy has cured to solid, you can sand the whole repair smooth, feathering the edges of the individual patches as you do so. After it is smooth, wash the repair well, rinse well, allow to dry and apply some paint to protect the epoxy from UV degredation.