Extremely Imbalanced and Gapped Dataset in Regression Problem

Currently I am working with a biological dataset with a range of 0-to-1 to do a multi-task regression with Deep Learning. However, this dataset has an empty gap in the range 0 to 0.2 (however there are several datapoints which are 0s) which can’t be interpolated due to biological reason. I also can’t randomly upsampling or downsampling my data with the same reason. In addition, there is also high tendency of a skewed data around 0.3. Until now I have only tried to train my model with standard HuberLoss and MSELoss and tried several type of models. However, I still don’t get “good” prediction result with correlation of only (~0.65). I came across several articles (such as this one) about weighted MSE but most of them talking about classification problem rather than regression. In addition, I read an article about imbalanced dataset in regression but seems like they did interpolation for the data.

My question: is it possible to improve my model by weightning the Loss function? If so how could I approach it?
Many thanks in advance!