What happens if you reduce the amount of sugar in a recipe. Will it affect the structure of the cake?

Depends on the recipe, and how much you reduce it.
Example: cookie recipe calls for 2 cups. You can reduce it by 1/4 cup without a problem.
Making icing: calls for 2 pounds of powdered sugar. It's possible to reduce it by as much as 1/2 cup but it's not going to change how sweetg many people find icing.

are you talking cakes or cookies?

It can affect the structure of the cake depending on how much you take out. Do you have a recipe you are thinking about. Can you post it here.

Generally if you keep the sugar to the same weight (not volume) as the flour, you are okay. But sugar is a tenderizer. If there are more toughening ingredients than tenderizing ingredients, you could get a tough cake, no so much a less sweet cake.