Experts and BHG readers answer.
moss instead of grass
Several conditions favor moss over grass. One of them is shade. If the area no longer is shady, but moss continues to overtake grass there, it may be due to excess moisture in the soil or possibly low pH. Is the area poorly drained or compacted? That could be what is holding moisture in. Aeration may help alleviate the problem. You might also take a soil test to confirm that the pH is in the correct range for grass. Most lawn grasses prefer a pH slightly on the acid side.
Creeping red fescue is one of the more shade tolerant grasses, and it might be a good choice to try in the spot.