Yes, BUT(!) keep in mind:
1) procedure can be different from one version to another
2) you can get a brick until you physically flush box
3) this is just a trick and no guarantee it'll repair your box

Okay, if you read this and understand all risks:
I have no R4 around, so how it work in S5

1) look at the shell script /www/pages/cgi-bin/upgrade_clear.cgi
2) scp downloaded_image.tar.gz to R4
3) tar zxf downloaded_image.tar.gz
if you can see runme.sh - you are lucky
4) sh runme.sh
in other case you should do many wood magic described in upgrade_clear.cgi
after
else
    if [ -e /dev/mmcblk0p3 ]; then
...
fi


I uploaded lighthttpd that built for your arch
https://www.dropbox.com/s/p0pypos1y8bqfsq/lighttpd.gz?dl=0

Download it separately on r4 and try to run
/your/folder/lighttpd -v