I’m not sure how you can save the shoes, short of taping them up but it’ll be a bodge. Your problem as you know if that you have a bit of plastic sticking into your heel and that’ll be hard to remove without compromising the structure of the shoe.
A few thoughts though:
It’ll likely have been caused by heel slippage. This can be remedied by using the lace-lock method of lacing them up, details here.
You can try taping your heel up with this tape, it’s really sticky and works well for this purpose, link here.
Finally, you could try different socks, perhaps a pair with 2 layers to prevent rubbing, something like this
Edit, and yes, look at how you wear them. Are they done up tight enough to stop movement? Do you slip them on/off without undoing the laces? Do they actually fit? To big a shoe might cause heel slippage. You can get through a pair in 2 months but you’d need to be putting some mileage in, and ASICS are normally quite hard wearing