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

Not tested this(Noctua NF-F12 iPPC-3000 PWM) on S9 but I'm not happy with them on S5:

Quite high temperature on low frequency, continuous work on 2880 rpm,  no magic. Maybe I'll test them with one big fan before go back to the original fans.

Look at the pool stats, I can't see the problem from my end.

Use the ps command to get the process id (PID) of this process.
kill -9 (PID)
But, I'm surprised that monitors/supervises lighttpd daemon can stop other process.
https://www.mankier.com/8/lighttpd-angel

Try to reboot, it'll restart all services
reboot

Closing ssh session will not kill daemon:)

Next time please fillow recommendation by @Not Fuzzy warm, it have more sence and less risky than patch application.

opkg is not Bitmain
https://wiki.openwrt.org/doc/techref/opkg
Upgrading packages is generally NOT recommended for most users, because a typical OpenWrt system stores the base system in a read-only SquashFS partition.

We can try at least there are no other way to solve it.

Nice question ) Possible but quite complex.

Can you run
/usr/sbin/lighttpd-angel -f /etc/lighttpd.conf

It provides some basic information about this binary.
Show me your:
opkg list-installed | grep [l]ight
and
opkg status lighttpd

Ok, seems the better way for you is just reflash Natasha with a newer firmware image. And it is recommended for all Ant devices as far as I know.

Heh, don't panic )
run these two commands
stat /usr/sbin/lighttpd
and
which lighttpd

There is external repository opkg and we can try to update lighttpd.

So far so good.
Run this please
/usr/sbin/lighttpd -v
and
/usr/sbin/lighttpd -tf /etc/lighttpd.conf

Hmm.. are you sure that your miner is mining?
I can't see process like this
{screen} SCREEN -S cgminer -t cgminer
but I have no Natasha here, just beaglebone.
edt. Ah bmminer here, ok.

Ok, look at log file: /www/logs/lighttpd.error.log
cat /www/logs/lighttpd.error.log

Can you show me disc space, it could be the reason:
df -k

No prob, I can help you.
If you use Windows, download http://www.putty.org/ and connect to antminer ip, it will open command line session.
ps | grep [h]tt will show you processes with name *htt*
maybe it'll be a help to show full process list:
ps
lighttpd is a web server that show you pages when you open ip port 80 in browser.
Hope it helps.

Try to ssh inside this box at least it pinged.
root/admin
Run `ps | grep [h]tt`
start it like
/usr/sbin/lighttpd -f /etc/lighttpd.conf
from command line.