Depending on the design of the study and the differences between the batches, removing the batch effect may or may not work. Suppose you have two experimental groups and two batches of samples, if the two variables are orthogonal then you have a better chance. You can assess the effect by principal component analysis, multidimensional scaling or clustering before after the removal of the batch effects. The goal should be to minimize the effect of the batch and still be able to extract true signal from the data.

So I guess the answer is yes and hopefully for the better :)

In short yes you should correct for Batches as that is part of the experimental design, as not correcting will produce unpredictable results. How you correct for that has different approaches. I prefer to correct batches as adjustment factors inside the model. If you want to read some technical problems due to not adjusting then read here one example: https://laplacebayes.wordpress.com/2018/03/22/large-effect-sizes-missing-information-produce-misleading-results/

Your other issues will be if you want to normalise the batches together or separately, and what effects that will have on the results.

See one example here https://github.com/uhkniazi/BRC_ILC_Joana_PID_21/blob/d5f541c7ac8ebc0a0250892581882e303c99aa82/09_exploratoryAnalysis.R#L129 that had strong batch effects. The workflow has examples where you can use simulations on the PCA components to compare models/decisions in order to guide choices.

Batch correction has been described from biologist point of view in this paper: Unbiased data analytics for biomarker discovery in precision medicine.

In brief, better to have some internal controls and use comBat algorithms. If you don't have internal control, there are some algorithm too. See the paper.