I think that most any camera in the $200-400 range will have trouble with night photos. Night photos are difficult - the key is more using a tripod and/or having a "fast" lens, which is only an option with a DSLR.
Having a camera that can do a higher ISO level with low image noise levels can help a lot, though. For this, you want a Point-n-Shoot with a larger sensor - 1/1.8" is the usual "big" PnS sensor. Don't worry about megapixels, they generally work against you when it comes to noise and most every current camera has more than enough.
Even if you want to use a flash, again, most PnSs don't do well here as they have a small flash with not much reach.
A pocket camera is more likely to have a tiny sensor and weak flash, so will often be worse than a larger PnS. This is only a generality, though, there are exceptions.
In other words... no matter what you don't, don't expect to suddenly be able to take terrific night photos. The best solution is to use a tripod, learn your camera's settings and how best to make them work for you, and take lots of pictures - you'll probably have some throwaways but hopefully you'll get some winners in there, too.