Whether correctly understanding when there can be an error calculation on MSE?

Question

Neural networks

Good afternoon everyone. I am a beginner in the knowledge of ANN and I would like to ask people who already know for help, namely, to check if I understand correctly when and how the MSE error is calculated?
This MSE formula is one of many loss functions and as I understand it is applied when using gradient descent?

The question itself: Can error calculation be done after batch and epoch? If after the batch we can get the error percentage and further modify the weights, then to get the error % for the batch, we just need to get the sum of the error from all elements of the training sample that are in the same batch?