There are several fixes to this, but my favorite is the ring flash. This is a flash that mounts to the front of your lens. Think crime scene photographers on CSI. This allows you to get extremely close to your subject and ensure your flash is directly on it. Furthermore, ring flashes are typically quite flattering for people, and the same is true for small critters. When the source of your flash is at the same angle as the front element of your lens, it removes shadows and illuminates texture and color. Another thing I like about the ring flash setup is that it has a small pilot light on the front to aid in focusing in the dark.
Another setup I use when wanting to create dramatic light angles and more shadows is a remote triggered flash. This is difficult at night, as it often requires an assistant to hold the flash for you, and a small light so that you can focus and compose your image. But, it can be very effective at creating some interesting angles.
My favorite technique is to use two light sources. The key, or primary, light is the ring flash. But, by using either a second flash slaved to the first, or a strong flashlight or head torch, I create a backlight to illuminate the subject’s edges.
Another simple fix for the lens shadow is to use a bracket system, but this will still require someone to help you focus by illuminating the subject until you get focus.
A word of caution, though, when using a flash at night on nocturnal animals. Their eyes are sensitive to light and a bright strobe can be blinding for them temporarily, making them vulnerable to predation after you’ve moved on. To attempt at countering this, I prefer the ring flash, because it is a lesser-powered strobe with a very short range. Also, I set it to 1-2 stops underexposed on the flash to reduce its intensity. I also attempt to bring the animal’s light sensitivity down by slowly acclimating it to a bright light using my head touch or flashlight before firing any strobes. This allows the animals’ pupils to contract before triggering the flash.
Settings
One of the other pitfalls early macro seekers will make is to shoot at a wide aperture. It seems like common sense that if you’re shooting at night that the iris of your lens needs to be wide open to allow in as much light as possible. However, all lenses have a similar characteristic, that the closer your subject is to the lens, the shallower the depth of field will be and the greater the fall-off towards the back of the image. So in the case of a 100mm macro lens, if I shot it wide open (f2.8) at its closest possible focal distance, the depth of field would be so shallow that the front of the eyeball would be sharp, but by the time it got to the eyelid, it would be soft.