It is almost certainly focus assist causing the delay. You mention seeing "a light" on the camera, which is what's used for focus assist when using the camera's built in, pop up flash. Try turning focus assist off in the menu, and see if that eliminates the delay (but in lower light you may see the autofocus struggle or fail to achieve focus... there may be a delay while the AF "hunts").
It's not "pre-flash". Pre-flash is a very quick, lower-power "pop" of the flash used to determine exposure, after which the main flash fires. This happens so quickly, and that first "pop" is so low powered (usually 1/64), that most people don't realize the flash is firing twice. You would not notice a delay.
Red-eye reduction is another thing.... usually a short, fast, bright strobing of the flash, prior to it firing normally. It will cause a delay, too... but the strobing on most cameras is pretty obvious. From the initial question, it sounded as if the "light" was steady, not strobiing.... which is more likely focus assist than red-eye reduction.
There's no delay with second curtain sync flash. So it can't be that, either. In digital 2nd curtain sync is done the same way it was done with film cameras... by timing the flash firing to the shutter closing, rather than the opening. 2nd curtain sync is used with slower shutter speeds and is primarily done so that "light trails" and "ghosting" end up appearing behind the moving subject. With standard 1st curtain sync those trails end up in front of the moving subject and look odd (subject appears to be going backward). This is an example of 2nd curtain sync (1/30 shutter speed)...
Not all flashes can do it, but High Speed Sync is sort of the opposite of 2nd curtain sync, in that it's used with especially fast shutter speeds. It makes possible to use flash with shutter speeds faster than the camera's flash sync, the flash duration is shortened in order to be completed during the shorter exposure, but this significantly reduces the distance the flash can reach. I would guess a D3200 has a 1/200 flash sync.... in order to use any faster shutter speed than that would require HSS (built-in camera flashes are so weak that it may not even be possible with them... a separate, more powerful auxiliary flash might be needed to do HSS).
Another possible delay while using flash might be noticed after the first flash exposure has made... waiting for the flash to recycle (recharge itself). The camera might prevent the shutter from releasing, until the flash has adequately recycled. Built-in camera flashes are often particularly slow recycling (and they also drain the camera's batteries fairly rapidly). More powerful accessory flashes with their own power supply (and possibly an auxiliary power supply as well) usually recycle a lot faster... Plus, thanks to their much greater light output, may only need to fire partially and not be discharged as fully. Auxiliary flashes also don't draw down the camera's battery.