Quantify the effects of a gene list between datasets

Question

[bioinf.questions]

I have two datasets of single cell RNA-Seq which contain mostly the same cell types, however differences are apparent between these two datasets (for example a clear separation on tSNE/UMAP before batch correction). I further have a list of several hundred genes of interest (some of which are differentially expressed between our datasets); we would like to quantify the effect/contribution of these genes on the difference between the datasets. I'd be very appreciative for any suggestions.

[bioinf.questions]

To be more precise our 2 "datasets" have been sequenced (and processed) together under the exact same conditions i.e there should be no technical batch effects. This is a key part of our study design. One set is effectively a model of the other and thus we want to identify the differences between them. They are clearly very similar, but we want to better quantify the differences.