If I entered a property through a cut hole in the fence with the purpose of cutting through the yard can I be charged with break@ enter?

No. This would be trespassing. B&E is when you break into a structure and enter it.

In Georgia it would just be tresspassing, but it will depend on how the state law in your state is written, the type of crime will depend upon the exact wording of your state law.

Here in Georgia, Breaking and Entering has to be a house, a car, a train.
Tresspass is entering the property, or not leaving a store when asked, or even theft under 500 dollars is called tresspass.