The circle of the peep should always be a little blurry. You should be focusing on the FRONT SIGHT PIN when you aim and prepare to release. You may be having trouble because your peep aperture is too small for low light. I had trouble with mine during a DART league, and went to a larger aperture. If you put the slightly blurry outer ring of the peep right on top of the in-focus front sight ring, as you should, and can't accomodate a slightly larger aperture hole,get a larger front sight, like a larger size HHA one. You are able to use a larger peep aperture that way and everything fits right in the sight alignment side of things.. I've also seen some experienced shooters carry a box of different sized peeps on the course, and actually swap out a couple times because of the hole being too small, or too large, for the light conditions. Larger apertures are absolutely required for early and late hunting under the trees. Nothing is worse that no shot because you can't see through a tiny aperture to shoot a game animal.
If you don't yet use one, get a Specialty Archery Hooded Peep. They are great.