A way to give more weight to particular data? Let us for these purposes say with are working with any feed forward neural network.

Let us also say, that we know beforehand that certain portion of our dataset arsignificantly more impactful or important to our underlying representation. Is there anyway to add that “weighting” to our data?
Yes, You just weight the entries of loss function like sum( Wi(li-t2)**2) for MSE. Most frameworks provides this ability. For example in keras fit method has a parameter - sample_weight: Optional Numpy array of weights for the training samples.