Use sudo tee with heredoc to append to existing file

You want the -a option to tee which appends rather than overwriting.

If you prefer to use cat, then go this route, first passing $OUTFILE as your output's receptacle:

$ OUTFILE=/path/to/restricted_write_access/file
$ sudo out=$OUTFILE sh -c 'cat << EOF >> $out
foo
bar
EOF'