What exactly do attached support units do in HQ?

[bloomstombs2]

And why cant they fight?

At first I thought they were there so that I can attach them to my divisions but I cannot seem to.
Example.

2 corps has 11 attached support units.

What do these guys do exactly? Shouldnt the HQ have a fight value?

[Timmeh]

HQ's have a CV of (zero) 0

Bring up your combat unit details to assign the SU's from the HQ

[heliodorus04]

You should be reading about HQ assignment of SUs during combat.

When an HQ has SUs assigned to it, rather than you manually assigning them out to combat units, then the SUs have a random probability of being assigned to hasty and deliberate attacks without any input from you.  Knowing the difference between how the probability assignment works for a hasty vs. deliberate attack matters a great deal.  I cannot remember whether it's in the combat section of the manual, or the HQ section of the manual, but it's probably a bit of both.

[gradenko2k]

This is going to be a little long, but prepare for a crash course in WITE combat (as I understand it):
The unit's Combat Value, or CV, represents the unit's ability to take and hold ground. The CV can also be used to get a rough estimate of how strong a unit is on the attack or defense, but the important thing to remember here is that the game actually models each individual unit shooting at other units.
That is, if you order a 100 CV infantry division to attack a 50 CV infantry division, they both shoot each other with their various squads and weapons, and their CV gets adjusted to reflect losses, leadership, changes in morale, etc.
The CV at the end of the fight, after all the shooting has concluded is then used to determine if the defender holds, retreats, routs, surrenders or shatters. The CV doesn't actually determine how effective an attack is - the shooting is still up to the actual guns. The CV just determines if you manage to take the hex or not.
Why is this distinction important? Because it explains why Support Units have a CV of 0, but can still dramatically change the course of a battle.
An artillery SU has a CV of 0, but when the shooting starts, its shots can take out a lot of enemy troops, and taking out those troops will then lower the enemy's final CV, increasing the chances of the final CV ratio being in your ratio, and therefore resulting in a victory.
===
Now for the game-mechanic aspects of using SUs:
If an SU is attached to an HQ, then there is a chance that the SU will participate in the combats of any of the HQ's subordinate units.

There is a limit of 6 SUs joining in a battle.
The chance of an SU joining a battle is based on the skill of the leader.
This chance gets a bonus if the HQ has not yet moved during the turn.
There's a range limit: The subordinate has to be within 5 hexes of the HQ for any HQ-attached SUs to participate
Hasty attack limit: If an HQ has moved within the turn, any SUs attached to it will no longer participate in any Hasty Attacks. This effectively means that any combat farther than 5 hexes away from your HQ can only enjoy SU participation if you use Deliberate Attacks, and again only if you move the HQ closer to the combat before engaging.
If you attach an SU directly to a combat unit instead of an HQ, then those SUs will always participate in combats involving its attached combat unit regardless of range or leader skill, but there's a limit of 3 SUs that can be attached.