Generate script in bash and save it to location requiring sudo

This is how I would do it:

sudo tee "$OUTFILE" > /dev/null <<'EOF'
foo
bar
EOF

Just putting sudo before cat doesn't work because >$OUTFILE attempts to open $OUTFILE in the current shell process, which is not running as root. You need the opening of that file to happen in a sudo-ed subprocess.

Here's one way to accomplish this:

sudo bash -c "cat >$OUTFILE" <<'EOF'
#!/bin/bash
#? [ ] / \ = + < > : ; " , * | 
#/ ? < > \ : * | ”
#Filename="z:"${$winFn//\//\\}
echo "This is a generated shell script."
App='eval wine "C:\Program Files\foxit\Foxit Reader.exe" "'$winFn'"'
$App
EOF

This starts a sub-shell under sudo, and opens $OUTFILE from that more privileged subprocess, and runs cat (as yet another privileged subprocess). Meanwhile, the (less privileged) parent process pipes the here-document to the sudo subprocess.