How can I give the cops presence in a mystery adventure?

[Mutazoia]

I'm running an urban murder mystery and, while I did expect the players to go to the guard for some things, they are actually being pretty forthcoming with information and requesting more guards to do their job. I sure didn't expect this from my experiences playing with them, though they've also participated in their own illegal activities in the campaign too.
Anyways, they've been giving the guards names and events of the suspects as they uncover information. Originally I hadn't had much more than some clues at the guardhouse (like a Speak with Dead transcript and autopsy report from a body that was later abducted). But with the frequent visits I feel the need to give the guard more presence. The PCs are speeding along the mystery at a decent clip, so I think it's fair to put the guards always one or two steps behind. But I don't want it to seem like they're completely ignoring the murders- they're understaffed, not malicious or ignorant.

How do I put more guards into this adventure, and even help the PCs, without having the guards simply solve everything for the PCs?

In your typical murder mystery, cops are usually the heavy handed idiots that are quick to leap to conclusions, arrest the wrong suspect and declare the case closed as quickly as possible. Having more cops on the case should actually be a hindrance, not help, for you players.

Usually, guards in a medieval/fantasy setting are not much more than grunts. They are there to be a physical deterrent to crime and not much more. They wouldn't typically have the intelligence or the inquisitive nature to be investigating anything beyond "what's in this pile of hay?" *poke it with a spear a few times*

You might have a few "higher ups", that would be the equivalent of an detective, but they are going to resent the PC's involvement (tell me how do do my job, eh? Think you can do it better than me, eh?) and will generally try to make the PC's life miserable while they investigate. They will try to be the first to a crime scene, make snap decisions based on little evidence (especially if the evidence is false or misleading) and try to keep the PC's from doing their own investigation as much as possible, even to the point of threatening to have them arrested if they keep "interfering".

Personally, I would suggest that the more they involve the guards, the more muddled things get as the half-witted guards trample evidence, rough up witnesses for no good reason, and generally try to get things over with as fast as possible, so they can return to their usual job of holding up a wall. The "detective" in charge of the case will be competent, but resentful of the PC's involvement and do their best to hinder the PCs movements, deny access to evidence and generally make their lives miserable, all while trying to beat them to the "solution" of the case.