Holes in lawn
It sounds as though your yard may have crayfish (or crawdads, as they are sometimes known). Crayfish invade lawns that are constantly wet, whether from a high water table near a body of water, as in your case, or from poor drainage or overwatering. They create holes about an inch in diameter, surrounded by a small mound of muddy soil. These mounds can make it a challenge to mow the wet site. There's no pesticide labeled for control of crayfish, so changing the environment to discourage them is your best bet. If you suspect that overwatering might be a factor, cut back on your irrigation. Poorly drained areas may be raised to improve drainage. But in your case, you might be better off rethinking what to grow in the wet area. Rather than trying to grow a lawn next to the creek, perhaps you could install a naturalized wetland or bog. Check with your local cooperative extension office for suggestions of native plants that will grow in wet sites. Then even if the crayfish make their mounds, mowing won't be an issue.