I try to use something stronger than cardboard, like an old piece of
drywall, or some screening if you have to bend it to get it through the
hole. I use plaster for my repairs, as I think it is stronger than mud.
If you put some plaster, mud, or even glue, on the edges of the patch
and the interior edges of the hole, it will hold very well when you
tighten the string, and give a good base for your repair coat.
I wouldn't worry too much about the strength of the patch, given that
you are dealing with a wall built of drywall, which is not noted for its
strength, and your patch will almost certainly be at least as strong as
the rest of the wall.
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