As Hillrider says, move the axle back to where the QR has a purchase on the totally unworr or undamaged part of the dropout. That's for starters, but also make sure to check the inside surfaces of the dropout for wear since most of the hold should come from the axle.

Now here's how to improve the bite. Get some medium/course sandpaper with an adhesive back. Cut donuts to fit over the axle faces. Clean the axle faces with acetone (not nail polish remover) to get a clean surface for good adhesion. press the sandpaper in place on the axle.

Now when the QR is clamped tightly the grit will bite into the inside of the dropout providing tons of holding power at reasonable clamping pressure.

As an alternate to the sandpaper, you can use beach sand, or commercial grit, and adhere it to the axle with nail polish or paint. This works best if you mix the grit into the paint and paint them on to a clean (acetone) axle face.

BTW- just in case, and please don't take offense, but are the QR springs on with the small end of the cones pointing in toward each other?