If exim is running but still cpanel emails about exim fails then do following….:

pico /etc/chkserv.d/exim
<supportLineBreakNewLine]—>

service[exim]=25,QUIT,554,/etc/rc.d/init.d/exim stop;/etc/rc.d/init.d/exim stop;/etc/rc.d/init.d/exim stop;/etc/rc.d/init.d/exim start

to:

service[exim]=25,QUIT,220,/etc/rc.d/init.d/exim stop;/etc/rc.d/init.d/exim stop;/etc/rc.d/init.d/exim stop;/etc/rc.d/init.d/exim start

Note: 554 -> 220 is the difference. Then run:
/etc/init.d/chkservd stop
/etc/init.d/chkservd start