There are a number of possible reasons for this.
Preparing the DEM for hydrological processing first is the key, rather than editting the results afterwards. ESRI provide a lot of information on the Hydro Tools. For instance, make sure you run the sink process to fill sinks before you do your analysis. Optionally, you can then use the fill tool to further normalise the DEM for hyrdoligical analysis. This latter tool also removes peaks which could cause other anomolies. Sinks and peaks are often glitches in the DEM data and will be your chief cause of poor flow accumulation.
Secondly, some of the areas you have highlighted could well be correct in that the flow is accumulating in the 'crease' between cells. Also, are you certain that there are no pools, ponds or lakes in your area?
I recomend that you follow the ESRI tutorials and undertake preparation of the DEM first.