Comment # 1 on bug 1004942 from Thomas Renninger

That is not the way it works.
You can easily get a binary included by adding a line:
inst_simple (or inst_multiple) cache_check ...

in a fitting (or new module if needed) in:
/usr/lib/dracut/modules.d/*/module-setup.sh

so that things work automatically.

For the first issue (dm-cache not added):
This module is included via 95dm dracut plugin.

/usr/lib/dracut/modules.d/90dm/module-setup.sh

You can explicitly add this via:
-a dm
or better:
add_dracutmodules+=" dm "
in /etc/dracut.conf.d/...

But this is also not how things should work (just work..).
The dm plugin is added via dependencies:

I expect you have an LVM on your root fs which is making use of some kind of
(dm-) cache feature?

Dracut should realize that you use an LVM on root in the check() function of:
/usr/lib/dracut/modules.d/90lvm/module-setup.sh

There is a dependency of lvm plugin to dm plugin (see depends() in
90lvm/module-setup.sh)

One source of the problem could be if you installed without caching (not sure
it's possible) and dm-cache is not loaded when mkinitrd is called. Then you
activate the stuff and reboot you are do not have the module included.

In this case you need to re-run mkinitrd if you change rootfs in a way that new
kernel modules or binaries are needed.

BTW: what/where is cache_check?