HDDs are about a thousand to over a hundred thousand times slower than RAM. The two main ways they can cause stuttering in games are:
■A pagefile on the HDD. Windows considers the pagesfile an extension of RAM, so gives pagefile accesses the highest priority and will freeze everything (even the mouse cursor) while it waits for the HDD to complete the pagefile read/write. Since you have a SSD, make sure the pagefile is on that and not your HDD. Don't bother trying to "save" your SSD from extra wear by moving the pagefile to the HDD. If that's really an issue, you need to add more RAM to your system.
■If you fill up your GPU's VRAM, the game will start dumping textures to make room. The vast majority of VRAM is used to hold textures "just in case" they will be used, so dumping some of them is the safest move. If the GPU has dumped a texture and suddenly needs it (you spin around in the game and something with an unloaded texture suddenly comes into view), the game will freeze while it waits to read the texture back from the HDD. The easiest fix for this is to turn texture quality down one notch. That'll drop texture VRAM use by 75%.
If neither of these work, the sure-fire way to determine if the HDD is the problem is to temporarily copy a game to the SSD, then disconnect the HDD and try playing the game off the SSD.